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Variations on interval graphs

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Variations on interval graphs
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Includes bibliographical references (leaves 144-147).
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by David E. Brown.

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Full Text
VARIATIONS ON INTERVAL GRAPHS
by
David E Brown
B.S., Metropolitan State College of Denver, 1999
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
2004

i,,.-


This thesis for the Doctor of Philosophy
degree by
David E Brown
has been approved
by
Ellen Gethner
Date


Brown, David E (Ph.D., Applied Mathematics)
Variations On Interval Graphs
Thesis directed by Professor J. Richard Lundgren
ABSTRACT
Interval graphs are a well-known type of intersection graph whose invention
or discovery is credited to Benzer in 1959, in the course of his studies of the
topology of the gene, and sometimes to Hajos in 1957, with respect to his purely
combinatorial question that asks basically which graphs have a representation by
a set of intervals so that vertices are adjacent if and only if their corresponding
intervals intersect. Interval graphs have been studied extensively since then
and many nice properties have been found to be associated with them; from
properties relating to structure to properties that admit much applicability.
Motivated by a problem loosely related to Benzers problem, Zhang created
probe interval graphs to model a specific problem in the physical mapping of
DNA. A graph is a probe interval graph if its vertices can be partitioned into
probes and nonprobes and intervals can be assigned to vertices so that vertices
are adjacent if and only if their corresponding intervals intersect and at least
one of the vertices is a probe. Whether or not the vertex partition is given
is a factor when determining if a given graph is a probe interval graph. In
1984 Harary, Kabell, and McMorris introduced interval bigraphs as a natural
m


extension of the interval graph idea. An interval bigraph is a bipartite graph
with an interval assigned to each vertex, and vertices are adjacent if and only if
their corresponding intervals intersect and belong to a distinct partite set. A talk
given by the author sparked a question that lead to the invention of interval k-
graphs. An interval k-graph is a fc-partite graph with an interval assigned to each
vertex with vertices adjacent if and only if their corresponding intervals intersect
and each vertex belongs to a distinct partite set; i.e., an interval bigraph is an
interval fc-graph with k = 2.
In this thesis it is shown that any probe interval graph is an interval /c-graph,
and this result is capitalized upon especially in the bipartite case. Specifically,
a necessary condition on the existence of a consecutively orderable biclique par-
tition is shown to hold for bipartite probe interval graphs. Also, the bipartite
probe interval graphs in which the probe/nonprobe partition can be made to
correspond to a bipartition are precisely the interval-point bigraphs and a list
of forbidden subgraphs is conjectured. An interval-point bigraph is an interval
bigraph in which the intervals corresponding to vertices of one of the partite
sets can be reduced to points. Interval point bigraphs (and the probe interval
graphs to which they correspond) are characterized by properties of their re-
duced adjacency matrices, as are unit interval bigraphs and interval bigraphs.
A unit interval bigraph is an interval bigraph in which all intervals have the
same length, a proper interval bigraph is an interval bigraph in which no inter-
val contains another properly. Unit interval bigraphs are characterized in several
ways. They turn out to be precisely the asteroidal triple-free bipartite graphs,
the bipartite cocomparability graphs (and hence the incomparability graphs of
IV


width two partial orders), bipartite permutation graphs, valuation bigraphs, and
are characterized further by an ordering of their vertices and by forbidden sub-
graphs. Some of these characterizations were found independently by Hell and
Huang, and some complement existing and newly found characterizations for
interval bigraphs.
Characterizations via consecutive orderings of certain subgraphs have been
given for interval graphs by Gilmore and Hoffman in 1965 and for probe interval
graphs by Zhang in 1994. In this thesis interval ^-graphs are characterized by
a consecutively orderable edge cover of complete multipartite subgraphs. In
particular a bipartite graph is an interval bigraph if and only if there is an
edge cover of bicliques that can be consecutively ordered. Whence the necessary
condition mentioned above for bipartite probe interval graphs.
Valuation k-graphs are to unit interval fc-graphs as indifference graphs are
to unit interval graphs. As a unit interval graph is the incomparability graph
of a semi-order, a unit interval /c-graph is the incomparability graph of a par-
tial order with width at most k. Unit probe interval graphs are shown to be
cocomparability graphs in several ways and hence they are the incomparability
graph of a partial order as well. Unit probe interval graphs are characterized
via forbidden subgraphs in the cycle-free case, and particular their relationship
with bipartite unit tolerance graphs is made precise.
A circular arc graph is the intersection graph of arcs of a circle. Closer in
age to interval graphs than any of the aforementioned classes of graphs, they
have also been studied extensively and characterized in many ways by A. Tucker,
in particular. Recently Hell and Huang have discovered that interval bigraphs
v


and proper interval bigraphs each correspond precisely to the complements of a
restricted class of circular arc graphs: e.g., proper interval bigraphs are precisely
the complements of proper circular arc graphs. We show a similar result for
interval point bigraphs, filling in the hitherto missing link in the circular arc
graph/complement of interval bigraph hierarchy.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed
vi


DEDICATION
I dedicate this thesis to my wife Tessa, and to my grandparents Addamae
and Dave Brown. They have put much more into me than I have into this
thesis.


ACKNOWLEDGMENT
The time I have spent at CU-Denver has been among the best of my life.
For this I have the following people to thank.
I am grateful to the Graduate Committee for providing me with teaching
assistantships, and to the faculty for the other honors that have been bestowed
upon me, including the Lynn Bateman Fellowship and the Ph.D. fellowship. Liz
and Marcia have all too often shown me what to do and when to do it, I thank
them for their patience and willingness to help.
I am grateful to Steve Flink for his company during some of the work that
created this thesis, and for his tutelage in many things non-mathematical. To
Rob Rostermundt and Steve Flink, thank you for the company and motivation
during the Christmas break that we spent preparing for the analysis prelim.
I am grateful to Dustin Stewart for the advice, mathematical and otherwise, he
has given me during the time we have shared an office together.
Thank you, Kathy, Bill, Stan, and Ellen for serving on my Ph.D. committee
and for the comments that have enhanced the quality of this thesis. And with
all my heart, I thank my advisor, Rich Lundgren. I know that to a great extent,
my experience at UCD has been so fantastic because of his advice, guidance,
and presence in general. I hope that in my future endeavors I can give to others
at least some of what he has given to me.


CONTENTS
Figures ............................................................... xii
1. Introduction.......................................................... 1
1.1 Prologue............................................................ 1
1.2 Notation............................................................ 2
1.3 Background.......................................................... 3
1.4 Evolution of my Research............................................ 5
1.4.1 Ordered Sets and Cocomparability Graphs......................... 8
1.4.2 (0, l)-matrices................................................... 13
1.4.3 Circular Arc Graphs............................................... 14
1.5 Progress on IBGs................................................... 15
1.5.1 T Restricted to Unit Intervals.................................... 16
1.6 Summary of Chapter Contents........................................ 17
2. Interval k-graphs and Probe Interval Graphs........................... 19
2.1 Definitions and a few details..................................... 19
2.2 A characterization for IkGs and a containment relationship......... 22
2.3 Some properties of IkGs............................................ 28
2.4 Forbidden substructures............................................ 30
2.5 Applications of IkGs............................................... 33
3. Interval Bigraphs................................................... 35
3.1 Background and definitions ........................................ 35
ix


3.2 Characterizations for IBGs.......................................... 39
3.3 Forbidden Substructures and Subgraphs............................... 50
3.3.1 Concluding Remarks................................................ 59
4. Interval Point Bigraphs............................................... 61
4.1 Background...................................................... 61
4.2 A Matrix Characterization........................................... 64
4.3 Forbidden Subgraph Conjecture....................................... 65
4.4 A Consecutive Order Characterization................................ 68
4.5 Probe Interval Graphs and Interval Point Bigraphs................... 73
4.6 Interval-Point Bigraphs and Circular Arc Graphs..................... 76
5. Unit Interval k-graphs ............................................... 82
5.1 Background.......................................................... 82
5.2 Characterizations................................................... 85
5.3 Proper IkGs........................................................ 106
6. Unit Probe Interval Graphs ........................................ 113
6.1 Background......................................................... 113
6.2 UPIGs are Cocomparability Graphs................................... 115
6.3 Cycle-free Unit PIGs............................................... 121
6.4 Bipartite UPIGs ................................................... 131
6.5 Concluding Remarks................................................. 133
7. Concluding Remarks and Future Directions........................... 135
7.1 Interval fc-graphs and Probe Interval Graphs....................... 135
7.2 Interval Bigraphs.................................................. 138
7.3 Interval Point Bigraphs............................................ 140
x


7.4 Unit Interval fc-graphs................................................. 140
7.5 Unit Probe Interval Graphs.............................................. 141
7.6 A Final Remark On Probe Interval Graphs............................... 142
References................................................................... 144
xi


FIGURES
Figure
2.1 An IkG that is not a PIG.............................................. 22
2.2 An IkG and its consecutive multipartite subgraphs..................... 27
2.3 The complement of a 6-cycle, which is not an IkG...................... 30
2.4 Examples of ATEs...................................................... 31
2.5 Two minimal forbidden graphs for IkGs................................. 32
3.1 H10 and an interval bigraph representation............................ 35
3.2 A zero partitionable matrix .......................................... 37
3.3 NL10: The smallest tree that is not an IBG and the only forbidden
subgraph for cycle-free IBGs......................................... 40
3.4 Interval bigraph representation for the matrix A...................... 47
3.5 An almost-insects interval bigraph representation.................... 48
3.6 A circular arc graph.................................................. 49
3.7 Forbidden configuration for Theorem 3.6 (e)........................... 49
3.8 Forbidden configurations for Theorem 3.6 (f).......................... 50
3.9 The finite families of finite sets that represent the forbidden comple-
ments of 2-clique circular arc graphs................................ 53
3.10 Some of the corresponding graphs of the families in Figure 3.9. The
thicker edges are ATEs or edge asteroids. The black vertices repre-
sent elements and the white vertices represent subsets.......... 54
3.11 The graphs corresponding to Tx, r2, r3............................ 55
Xll


/
3.12 The three insects. ................................................... 58
3.13 The bugs........................................................... 58
3.14 The exobiclique of a bug. Black vertices represent X............... 59
4.1 Three IPBGs, dark vertices belong to the point partition........... 63
4.2 Forbidden X-consecutive bigraphs................................... 69
4.3 IPBGs from Tuckers list........................................... 70
4.4 Forbidden interval point bigraphs; dashed edges may or may not be
present............................................................. 71
4.5 Example of a bipartite PIG with P,N partition not a bipartition. . 74
4.6 H10 has a consecutive partition into stars, but is not a probe interval
graph.............................................................. 75
4.7 A circular arc representation of the complement of an interval point
bigraph using the method in the proof of Theorem 4.10. The dashed
lines are meant to indicate the arcs............................. 80
5.1 A UlkG in which the ability to represent it rests on using a particular
interval class assignment. The horizontal dashed lines distinguish the
interval classes (as do their shadings), the vertical dotted lines serve
to help determine intersection..................................... 84
5.2 Forbidden unit interval bigraphs, include C*,, k > 6............... 91
5.3 Forced adjacency diagram for Theorem 5.16: dashed edges are forced 100
5.4 The labeling satisfies (i), but not (i) and (ii) of Theorem 5.16 . . . 100
5.5 Forbidden UlkGs that are asteroidal triple-free................... 107
xiii


5.6 A function representation for a proper interval 3-graph ala Theorem
5.23. Directly below H is its proper interval /^-representation, and
directly below that is a function representation for H......... Ill
5.7 A function representation for F: an interval k-graph that is not a
proper IkG......................................................... 112
5.8 A function representation, and a permutation representation for a
proper interval bigraph.............................................. 112
6.1 NC7 is a tolerance graph, not a bounded tolerance graph............... 118
6.2 A unit tolerance graph that is not a PIG.............................. 119
6.3 A parallelogram graph................................................. 121
6.4 A parallelogram representation for a bounded tolerance graph. . 121
6.5 Forbidden partitioned cycle-free UPIGs darkened vertices are probes. 122
6.6 Labeling used in lemmas 3.1, and 3.2.................................. 122
6.7 Forbidden PPIGs and UPIGs............................................. 123
6.8 Cases for an unsigned vertex v reducible vertices are labeled r,
darkened vertices have been assigned to P.......................... 130
6.9 Tj and 7Tj............................................................ 130
6.10 Forbidden bipartite UPIGs (include Fi,F2 and H, i > 0)............... 131
7.1 A parallelogram representation for a weakly proper PIG................ 138
7.2 2-trees that are not UPIGs............................................ 142
7.3 Possible subgraphs induced on probes of G of Figure 7.2............... 142
7.4 Asteroidal triple-free, 3-chromatic, minimal, forbidden PIGs. The
principle for constructing these can be used to create infinitely many
minimal forbidden PIGs......................................... 143
xiv


1. Introduction
1.1 Prologue
A common problem in combinatorics is the following. Given a structural
property X, determine a collection or list of objects £ such that, for each object
of £, the presence of the object precludes X, and if every object of £ is absent,
then X is present. A list £ is complete relative to X, if £ is minimal as a list
and the absence of every item on the list guarantees that X exists. The work in
this thesis stems from the attempt to determine an £ for varieties of intersection
graphs (serving as A) that have a family of intervals of the real line as their
representation. Furthermore, whenever possible, we attempt to construct £ so
that it consists of (sub)graphs. If such an £ is determined, we have a forbidden
induced subgraph characterization. Such a characterization is difficult when the
list is extensive, and typically the determination of the list may only arise from
ad hoc arguments. But some of the graph classes we consider turn out to have
nice characterizations. By nice we mean that their corresponding £ is short
or easy to describe. Given a particular X, and a possibly complete list £' of
forbidden substructures, it is relatively easy to see that the prohibition of every
thing in £' is necessary, but proving that £' is complete is typically difficult,
and one is usually left with proof techniques that fall short of elegance.
Given some combinatorial object O and structural property A, the recogni-
tion problem is that of determining whether O has X. The certification problem]
relative to O and X, is that of proving that O does not have X. Assuming that
1


X is important, the motivation for finding £ relative to X, lies partly in the
fact that £ makes the recognition problem easy. Because, even if £ is large as a
list and its items are complicated, the complexity of its items is relative to the
size of O for which X is being searched. Therefore, almost always, having £ for
X leads to fast recognition algorithms. If £ is not conducive to fast recognition
algorithms, it may be conducive to fast certification algorithms; this is because
if £ is known for a particular X, one need only find an object from £ in O to
certify that O does not have X.
1.2 Notation
Notation will be developed as needed, but at the outset we can say the
following (now typical) notational conventions will be used. A graph G consists
of a set of vertices V (or V(G)) and a collection of edges E (or E(G)) which in
turn consists of distinct unordered pairs of distinct elements of V. Note: most
authors call this a simple graph, we call it simply a graph. Some authors would
say that a graph G is an irreflexive, symmetric relation E on V. To denote
an edge we juxtapose two vertices and say the vertices are adjacent; that is,
for u,v e V, uv e E denotes u and v are adjacent or uv is an edge. Also,
we may use the symbol -o- between vertices to indicate that they are adjacent
and / to indicate that they are not. Given a subset V' C V(G), the graph
induced on V' is denoted G(V') and is the subgraph in which vertices of V'
are adjacent in G(V') whenever they are adjacent in G. A graph will often
be denoted by an ordered pair that indicates both the vertex and edge set:
G iy,E). A graph G (V,E) is k-partite if V can be partitioned into k
subsets V = Vj U U Vk such that each edge consists of vertices from distinct
2


partite sets of V, alternatively, G(Vi) has no edges for each i. A subset of
vertices V' C V{G) with G(V') having no edges is called an independent set.
Much of this thesis is devoted to bipartite graphs. So, in case G is bipartite
and V partitions into V = X U Y, such a graph is denoted G = (X,Y,E).
A complete graph is one with every pair of vertices adjacent, while a complete
multipartite graph is a fc-partite graph with every pair of vertices that belong to
different partite sets are adjacent. A directed graph or digraph D = (V, R) is a
generalization of a graph in which R, the set of arcs consists of ordered pairs of
V, and for vertices u and v if the ordered pair (u, v) G R, then we may denote
this u > v. So, a digraph D = (V,R) is simply a relation R on V with no
restrictions; that is, the relation could be reflexive for some elements of V and
not others, and (u, v) G R does not imply (v,u) G R.
It is typical to represent a graph with a drawing in which vertices are de-
picted by dots and if two vertices are adjacent, then a line connecting them is
drawn. A digraph is typically represented in the same way, but with arrows
indicating the order; in a drawing for a digraph D = (V, R) with (u, v) G R, an
arrow is drawn from u to v.
In this thesis, vertices often correspond to objects, like intervals, and the
correspondence will be indicated by subscripts or by functional notation. For
example, I(v) may denote the interval that corresponds to vertex v, so may Iv,
and if needed, l(v),r(v) denote the left and right endpoint of I(v).
1.3 Background
An intersection graph is a graph G whose vertices correspond to a family
of sets T and vertices are adjacent if and only if their corresponding sets in-
3


tersect; T is called a representation of G when G is the intersection graph of
T. Any graph is the intersection graph of some family of sets, see [33]. But
when T is restricted or the rules that determine adjacency are modified, in-
teresting classes of graphs may be created. The recent text [34] evinces the
diversity and importance of the intersection graph perspective. Probably the
most well-known of the intersection graphs is obtained when T is restricted to
be intervals of M. In this case, the intersection graph of T is an interval graph.
Interval graphs have been extensively studied and characterized in many ways,
including a characterization by forbidden substructures. The papers [4], [13],
and [15] are the seminal papers on interval graphs and the techniques and char-
acterizations in them are easily seen to be the prescient ones for many of the
analogous intersection graph classes that are chic now. The texts [34], [20], and
[40] contain excellent introductions and exhibit refinements of the proofs in the
aforementioned papers.
The interest in many classes of intersection graphs are application driven.
For example, interval graphs are often attributed to an attempt to model ideas
about the fine structure of the gene, see [1]. Advances in the field of molecular
biology, and genetics in particular, solicited the need for a new model. In 1994
P. Zhang introduced probe interval graphs in an attempt to aid a problem called
cosmid contig mapping, a particular component of the physical mapping of DNA,
see [48, 49]. A probe interval graph (PIG) is a graph G = (V, E), where V
can be partitioned into (P, N), such that there is an interval corresponding to
each vertex and vertices are adjacent if and only if their corresponding intervals
intersect and at least one of the vertices belongs to P. PIGs generalize interval
4


graphs (put N = 0) and are an instance of an intersection graph with a modified
adjacency rule.
We note that although applications often accompany mathematics as mo-
tivation for study, sometimes it is clear that the mathematics has strayed far
from its conception as a consequence of application and is now studied for its
own sake. This is what happens in this thesis. For example, interval graphs can
also be attributed, and perhaps more appropriately so, to a question posed by
Hajos in 1957, see [21]. He asked, basically, what graphs have a representation
by collection of intervals of M. Benzer never mentioned interval graphs as such,
but the mathematical community picked up and ran with the idea. Most of the
research in this thesis is driven by combinatorial questions fueled by the modi-
fication of a definition or the attempt to find a result analogous to a canonical
one. The next section tells the story of how some of the results came to be,
approximately.
1.4 Evolution of my Research
Most of the work in this thesis stems from an attempt to characterize PIGs.
Because of their purported application to physical mapping of DNA, combined
with the fact that they generalize interval graphs, the research community has
shown an interest in PIGs. While investigating PIGs, Cary Miller, for reasons
having to do with the biology of the application PIGs model, asked: what if
at least is changed to exactly in the definition of a PIG? Our preliminary
investigation showed that a distinct and interesting class of graphs is obtained
from this definition. We eventually came to call this class of bipartite graphs
interval bigraphs (IBGs).
5


Due mainly to nomenclature, for quite some time we did not realize that
IBGs had already been introduced and studied. While we saw IBGs as a natural
extension of PIGs, IBGs were introduced in 1984 as a natural extension of
interval graphs and called bi-interval graphs, see [22]. There was also an analogue
to interval bigraphs in the context of digraphs introduced in 1989, see [10]. An
interval digraph (ID) is a directed graph with an ordered pair of intervals (Su, Tu)
corresponding to each vertex u such that u > v if and only if SU(~\TV ^ 0. In
[10] IDs are introduced as an analogue to interval graphs. Das et al. [10]
also seemed to be unaware of the paper [22] that introduced and (erroneously)
characterized bi-interval graphs. We note that the models for both IDs and IBGs
are essentially the same: let each Sx correspond to a vertex x G X and each
Ty correspond to a vertex y G Y and you will get an IBG G = (X, Y, E) with
\X\ = \Y\. Therefore, results obtained from the perspective of interval bigraphs
complement those for interval digraphs. We note, with an admitted bias, that
we believe the bipartite graph perspective to be more fruitful because of the
structure present in bipartite graphs that is not present in directed graphs.
An alternate definition of IBGs is the following: An IBG is the intersection
graph of two different families of intervals with vertices adjacent if and only
if their corresponding intervals intersect and belong to different families. Note
that an IBG is bipartite with each partite set corresponding to one of the two
interval families. At a talk given at DIM ACS by the author in the summer
of 2000 on early results on IBGs, T.S. Michael asked the following question:
Given a graph, how many different interval classes are required to represent it
so that vertices are adjacent if and only if their corresponding intervals intersect
6


and belong to distinct classes?. Whence interval k-graphs (IkGs); i.e., IBGs
with two replaced by A. The answer to the question for some graph G,
by the way, is x{G), but not all graphs can be represented in such a way. For
example only a very restricted class of trees are IkGs, and an n-cycle, for n > 5
is not an IkG for any k. In Chapter 2 we prove that IkGs generalize PIGs, and
are perfect, and we give a characterization for IkGs in terms of a consecutively
orderable collection of subgraphs. However most of this thesis is focused on
IBGs and subclasses of IBGs obtained by restrictions placed on one or both of
the classes of intervals in the representation. In particular, at various times, and
for various reasons, we decided to investigate the consequences of restricting all
intervals to unit length, giving the unit IBGs. At another time, we investigated
what happens when one interval class corresponds to points only, giving the
interval point bigraphs. We obtain very conclusive results for unit IBGs, see
Chapter 5, and for interval point bigraphs, see Chapter 4. But for IkGs (k > 3),
even the case in which all intervals are of unit length remains open. We remark
that, because of the structure imposed upon the representation, the unit case
is often the most fruitful in terms of discovering structural properties. Indeed,
unit IBGs turn out to be identical to many other well-known classes of bipartite
graphs and the complements of others, see Chapter 5.
Remark: The author contends that IkGs have more structure than PIGs and
so it makes sense to investigate IkGs primarily, with an eye open for ancillary
results regarding PIGs. This is the approach taken here and it has turned out
to be fruitful in some cases.
Many of the directions for investigations in this thesis are due to the work by
7


earlier mathematicians who studied interval graphs and probe interval graphs
and from contemporary ones studying interval bigraphs and interval digraphs.
The results of Roberts, Gilmore and Hoffman, Hell and Huang, and Das, San,
Sen, Roy, and West on unit interval graphs, interval graphs, interval bigraphs
and circular arc graphs, and interval digraphs, respectively, have sparked many
of the ideas that led to results. We thank them for their passive, but powerful
contribution. The next paragraph gives some specifics to how the aforemen-
tioned mathematicians have motivated this research.
The indifference function Roberts associated with unit interval graphs, see
[40, 38], led to the idea of valuation functions in relation to interval k-graphs.
The consecutive order characterizations of Gilmore and Hoffman in [15] led us
to ask, whenever plausible, whether such a characterization exists for the class
under consideration (this will become obvious). The vertex ordering character-
izations by Hell and Huang motivated a result in Chapter 5. Hell and Huangs
results relating interval bigraphs to circular arc graphs in [24] motivated the
characterization of interval point bigraphs in Chapter 4, identifying them with
a restricted class of circular arc graphs. Of course, we have used results of many
others directly and they are cited. In particular, A. Tuckers work on (0,1)-
matrices is cited more than once in Chapter 4, and Li Shengs work on PIGs
has been useful; in particular, her characterization by forbidden subgraphs for
PIGs that are cycle-free (see [43]) suggested the work in Chapter 6.
In the subsections that follow, we discuss some of the other mathematical
things which have come to bear, motivate, or relate to results in this thesis.
1.4.1 Ordered Sets and Cocomparability Graphs
8


An ordered set V = (X, <) consists of a set X and a binary relation < on X
that is irreflexive (x it x for all x £ X), transitive (x < y and y < z ==> x < z
for all x,y,z £ X), and therefore asymmetric (x < y ==£ y it x for all
x,y £ X). We say that elements x,y of X are comparable if either x < y or
y < x, and otherwise they are incomparable. A partially ordered set (poset)
V = (X, <) has relation < that is reflexive (x < x for all x £ X), antisymmetric
(x < y and y < x => x = y), and transitive.
There are two common or natural ways to associate a graph G = (X, E)
with an ordered set V = {X, <). One way is to define the edges thus: E =
{xy : x < y or y < x}] this renders G the comparability graph of V. The
incomparability graph of V has edges defined by E = {xy -.x cocomparability graph G is one in which the edges of G can be directed so that
the resulting digraph is transitive. So, a cocomparability graph G = (V, E) has
an ordered set on V in its complement G defined by any transitive orientation:
make x < y whenever x y y in G. In this thesis, we have asked ourselves
for every class of graphs we investigate Does this class belong to the class of
cocomparability graphs? When the answer is affirmative, there is an order
defined by the complement of any graph in that class. This order may be
interesting fodder for future study. But since we care only about the orders that
are defined by orienting the complements of graphs, that is, by giving each edge
a unique direction, the relation created can be thought of as antisymmetric.
Also, we concern ourselves only with loopless graphs, so we may either assume
that in the complement there is understood to be a loop at every vertex; or we
may assume the opposite and find that, for our purposes, either assumption is
9


of no consequence. Thus, we could regard the order on the complement of a
cocomparability graph as a poset. And since this, for the most part, is what the
prominent authors in the literature do, we will do the same, see [18, 19, 3].
Remark: There is actually discrepancy among some authors and even among
the same authors from paper to paper regarding their definition of a partially
ordered set. But the discrepancies stem from pragmatic reasons, and are irrel-
evant in terms of the theory of partially ordered sets; that is, the well-known
results from poset theory still apply given any of the different definitions that
the author has seen.
Here are some other terms we will use. A poset of width k decomposes into
k disjoint linear orders; a linear order in a poset (with the conventions discussed
above) is one with no incomparabilities among different elements. For example
we can define a width k poset on the complement of a k-partite cocomparability
graph G, since in G there will be k complete subgraphs, and a complete subgraph
with a transitive orientation defines a linear order.
The following example will illustrate how posets relate to interval graphs.
In an attempt to model the non-transitivity of indifference, Luce developed
a model for preference motivated by the concept of threshold in psychology, see
[32]. He contended that, for a set of things X and the preference relation R,
one seek a function / : X > R and a just-noticeable-tolerance 5 > 0 with xRy
(x is preferred to y) if and only if f(x) > f(y) + 5; i.e., if the value placed
on x is sufficiently larger than the value placed on y. This representation led
to the development of an order called a semi-order. A semi-order < on X is
a binary relation with the following properties: For all x, y,z,w X, (1) <
10


is irreflexive, (2) x < y and z < w imply x < w or z < y, (3) x < y and
y < z imply x < w or w < z. Scott and Suppes later characterized semi-orders
via the existence of a function / and a tolerance <5 as above, see [42]. In his
Ph.D. thesis, Roberts studied graphs with adjacency determined by the rule
(A): vertices u and v are adjacent if and only if \f(u) f{v)\ < 5. Now, observe
that a preference relation represented by / gives rise to a transitive digraph, now
make this digraph a graph by including a reversed copy of each arc, and finally
take the complement of this graph. What you get is a graph whose adjacency
is determined by (A), and it turns out, an interval graph in which all intervals
have length <5 the unit interval graphs.
We have observed that the complements of unit interval graphs can represent
preference relations, whence unit interval graphs are also called indifference
graphs, and they are the incomparability graphs of semi-orders.
We extend the idea of indifference graphs, without the psychology appli-
cation, to IkGs. Given 5 > 0, define k functions : Xi > R (1 < i < k),
and a ^-partite graph G = (A\, ...,Xk,E) such that uv G E if and only if
\fi(u) fj(v)\ < 5, u £ Xi,v G Xj and i / j. It is easy to see that the graph G
is an IkG where all intervals are of length 5: a unit interval k-graph (UlkG), or
a valuation k-graph, with ff a valuation function. In Chapter 5 we show UlkGs
are cocomparability graphs in 2 ways and hence they are incomparability graphs
of some poset of width k. This is because, as we have remarked above, any IkG
is fc-colorable and hence has k independent sets which in turn form complete
graphs in the complement, and a complete graph yields a linear order. The
structure of the corresponding posets and of UlkGs is an open problem, except
11


when k 2, the unit interval bigraphs (UIBGs). UIBGs are precisely the in-
comparability graphs of a poset of width 2, and equivalent to many other classes
of graphs and are characterized by a variety of structural properties. Motivation
driving the model for UlkGs includes modeling the simultaneous rankings of not
necessarily distinct sets, the s, by multiple judges, modeled by the fis. Un-
der this interpretation, a transitive orientation of a UlkG models the preference
structure on each partite set, and if the valuation functions are constructed with
respect to some uniform scale, then the indifference and preference structure is
captured in the graph and complement, respectively. The paragraph preceding
this one is intended to suggest potential applicability of UlkGs, perhaps in a
way similar to unit interval graphs.
A unit probe interval graph (UPIG) is a PIG in which all intervals can be
made to have the same length in some representation. In Chapter 6 UPIGs
are shown to be cocomparability graphs. Thus, by taking the complements
of the bipartite graphs, that we conjecture to be a complete list of forbidden
subgraphs for bipartite UPIGs, one obtains the forbidden sub-posets that would
serve as a forbidden sub-poset characterization for the width-two partial orders
that correspond to bipartite PIGs (provided the conjecture is true). The partial
orders corresponding to UPIGs generalize semi-orders similar to the way PIGs
generalize interval graphs. More specifically, the nonprobes form an independent
set by definition, and hence a complete subgraph in the complement. Any
complete graph has a complete transitive order, that is, a linear order. Hence the
partial orders corresponding to UPIGs are an interplay of a semi-order induced
on the probes, and a linear order induced on the nonprobes. These orders have
12


not been studied. Perhaps there is an application in psychology similar to the
one for semi-orders and unit interval graphs.
1.4.2 (0, l)-matrices
Any m x n (0, l)-matrix corresponds to a bipartite graph G = (X, Y, E) with
|X| = m and [Y| = n; and conversely, with A(G) denoting the (0, l)-matrix
obtained from G with entry (i,j) = 1 if and only if and yj are adjacent in
G. A (0, l)-matrix has a zero-partition (ZP) if there are independent row and
column permutations after which the zeros may be labeled R or C so that to
the right of every R is an R and below every G is a C. A (0, l)-matrix has
the consecutive 1 s property for rows (C1R) if the columns can be permuted so
that the ls in every row appear consecutively; the consecutive ls property for
columns (ClC) is defined similarly. A (0, l)-matrix has a monotone consecutive
arrangement (MCA) if it has a zero partition with the additional property that
below and to the left of every C is a C and above and to the right of every R is
an R. Let M be a (0, l)-matrix; then if M has an MCA, it has C1R and ClC; if
M has C1R or ClC, then it has a ZP. These containments are proper and each
(0, l)-matrix property characterizes a class of IBGs. If G is an IBG, then A(G)
has a ZP, and conversely (see Chapter 3). Recall that if one of the families of
intervals representing an IBG G can be restricted to be points, then we call G
an interval-point bigraph. If A(G) has ClC or C1R, then G is an interval point
bigraph and conversely (see Chapter 4). If A(G) has an MCA, then G is a UIBG
(or belongs to any of the equivalent classes), and conversely (see Chapter 5). In
[45], A. Tucker gave what he called a structure theorem for ClC: he determined
a minimal forbidden list of bipartite graphs whose corresponding (0, l)-matrices
13


do not have ClC. We conjecture such a list for a (0,l)-matrices having ClC or
C1R. The forbidden subgraph characterization for UIBGs in Chapter 5 gives a
structure theorem for the (0, l)-matrices with an MCA. If IBGs are characterized
in terms of forbidden substructures, then we would have a structure theorem for
the zero partitionable (0,l)-matrices.
1.4.3 Circular Arc Graphs
A circular arc graph is the intersection graph of arcs of a circle. We call
the circle from which arcs are obtained the host circle. Circular arc graphs have
been extensively studied; at the time this thesis is written, a popular scientific
internet search engine gives over 50 articles on circular arc graphs. The seminal
papers are probably [46] and [44] by Tucker, while a nice introduction and
further references and applications can be found in the texts [20] by Golumbic
and [40] by Roberts. A 2-clique graph is a graph in which the vertices can be
partitioned into two cliques so that all edges not belonging to the two cliques
lie between them; a bipartite graphs complement is a 2-clique graph. The 2-
clique circular arc graphs are characterized by a list of bipartite graphs that
cannot occur in their complements in [36]. In [12] Feder, Hell, and Huang
give a different characterization of 2-clique circular arc graphs via a forbidden
substructure that captures the prohibitory property in all the graphs derived
from [36]. This new characterization probably inspired the connection between
interval bigraphs and a restricted class of 2-clique circular graphs proved in
[24]. Also in the same work, proper interval bigraphs are shown to be precisely
the complements of proper circular arc graphs. In [44] it is shown that proper
circular arc graphs are necessarily 2-clique graphs. Capitalizing on this trend,
14


we prove that interval point bigraphs are identical to the 2-clique circular arc
graphs in which the vertices of one of the covering cliques can be labeled in a
circularly consecutive fashion.
The next two sections give more thorough summaries and introductions to
the meatiest chapters.
1.5 Progress on IBGs
We discuss the content of Chapter 3. As this thesis is written, there is no
forbidden subgraph characterization for interval bigraphs (henceforth IBGs) in
general. Although [22] purports to have such a characterization, it does not. We
give a complete characterization for the IBGs that are cycle-free in Theorem 3.3.
But there are several ways, other than by forbidden subgraphs, to characterize
IBGs at this point.
P.Hell and J. Huang showed that IBGs are precisely the complements of
2-clique circular arc graphs in which no two arcs cover the host circle, and are
characterized by two different orderings of the vertices, see [24], In [10] interval
digraphs are characterized via the structure of their adjacency matrices, and
we do the same for IBGs (in terms of their reduced adjacency matrices). A
collection of subgraphs Q = {G?} is consecutively ordered if v Gi D Gk implies
v £ Gj for any vertex v and i < j < k. We also characterize IBGs via an
edge cover of bicliques that can be consecutively ordered, motivated by various
consecutive order characterizations for interval-type intersection graphs in the
literature. A new construction for a representation of an IBG via its reduced
adjacency matrix is given.
An asteroidal triple (AT) of a graph G is a set of three vertices such that
15


there is a path between any pair that avoids the neighborhood of the third. In
[4] interval graphs are characterized as those chordal graphs that have no AT
in any induced subgraph. An IBG analogue for an AT is an asteroidal triple
of edges (ATE): a set of three edges such that there is a path between any two
that avoids the neighborhood of the third edge, where if e = uv G E, then
N(e) = N(u) U N(v). In Chapter 2 it is shown that an ATE is a forbidden
substructure for a graph to be an interval /c-graph, and hence an IBG, but the
converse is not true.
In [37] it is conjectured that G is an IBG if and only if it has no ATE
and no insect (a class of graphs defined in Chapter 3), see Figure 3.12. But in
[24] this conjecture is refuted, the list of insects is extended, and the modified
would-be conjecture is purported to be refuted by undisclosed examples. At
the Rocky Mountain Discrete Math Days, P. Hell claimed that the examples
he and J. Huang are large. Also in Chapter 3, we summarize what is known,
and what can be deduced from the literature, regarding the forbidden subgraph
characterization for interval bigraphs.
1.5.1 T Restricted to Unit Intervals
While E with no restrictions renders the characterization problem for probe
interval graphs and interval /c-graphs difficult, if T is restricted to consist of unit
length intervals, giving UPIGs and UlkGs, the problem becomes more tractable.
It turns out that a surprisingly lengthy list of characterizations for unit interval
bigraphs can be made when the various results are put together; such a list is
developed in Chapter 5.
A graph G = (V, E) is a tolerance graph if there is a function t : V > R
16


and a collection of intervals {/}, Iv corresponding to v G V such that uv G E
if and only if \IV fl Iu\ > min{t(u), t(v)}. G is a unit tolerance graph (UTG) if
it is a tolerance graph and \IV\ = 1 for each v V. Any probe interval graph
is a tolerance graph. To see this, take a probe interval graph G = (P,N,E),
put t(v) = max^evd/yl}, and t(u) = e, for u G N, where e is an arbitrarily
small positive constant. Also any UPIG is a UTG, but not conversely. In
Chapter 6 UPIGs are characterized in the cycle-free case and in the bipartite
case a characterization is conjectured that is substantiated by their relationship
to UTGs. Specifically, the results regarding bipartite UTGs in [3] substantiate
the conjecture.
UTGs have interesting alternative representations. A parallelogram graph
is the intersection graph of parallelograms in which two edges lie on parallel
lines. In [2] it is shown that G is a UTG if and only if it is a parallelogram
graph. Therefore, UPIGs belong to the class of parallelogram graphs. This fact
yields an automatic proof that UPIGs are cocomparability graphs, and suggests
that their containment in the class of parallelogram graphs could be further
capitalized upon.
1.6 Summary of Chapter Contents
In Chapter 2 the stage is set with the introduction of interval ^-graphs.
They are shown to be a class of perfect graphs, to be a generalization of probe
interval graphs, and are characterized by the presence of consecutive subgraphs.
Chapter 3 focuses on interval bigraphs. They are characterized via their reduced
adjacency matrices and a summary of known characterizations and forbidden
substructures is given. In Chapter 5 unit interval bigraphs are characterized in
17


over a dozen ways, including a forbidden subgraph characterization, a reduced
adjacency matrix characterization, and a vertex ordering characterization. Also,
following Roberts, unit interval bigraphs are shown to be equivalent to proper in-
terval bigraphs. A consequence of the characterization for unit interval bigraphs
is a structure theorem for the (0,1)-matrices with a monotone consecutive ar-
rangement. Other consequences are, of course, that a surprisingly large list of
other mathematical objects have a myriad of ways to be represented. Interval
point bigraphs are characterized via their reduced adjacency matrices, via con-
secutively orderable subgraphs, and shown to be equivalent to a restricted class
of probe interval graphs and a restricted class of circular arc graphs in Chapter
4. We make a step toward a forbidden subgraph characterization for unit probe
interval graphs by giving a complete list of forbidden subgraphs for cycle-free
UPIGs and conjecture a complete list for unit probe interval graphs in the bi-
partite case. In the concluding chapter, the open problems are summarized, and
future directions for research are discussed.
18


2. Interval /c-graphs and Probe Interval Graphs
In this chapter, we introduce our largest class of graphs, interval A;-graphs.
We give a characterization via consecutively orderable subgraphs, show that they
generalize probe interval graphs, and give two proofs that they are perfect. We
define and prove a forbidden substructure for interval fc-graphs that is analogous
to an asteroidal triple, and, because of the containment relationship we show,
is also a forbidden substructure for probe interval graphs. We end the chapter
with a couple of applications.
2.1 Definitions and a few details
A proper coloring of a graph G is an assignment of vertices to colors so that
adjacent vertices have different colors. So, with G a graph, and / : V(G) >
{1,..., k} the function that assigns a color to each vertex, if vertices u and v are
adjacent, then f(u) ^ f{v). The subset V, C V{G) defined by Vi = {v e V(G) :
f(v) = i} is called color class i; by definition Vi is an independent set in G, that
is, G(Vi) has no edges. A graph G with a proper coloring is an interval k-graph
(IkG) if each vertex v can be assigned an interval I(v) of R such that vertices
are adjacent if and only if their corresponding intervals intersect and each vertex
has a different color. The collection of intervals {I(v) : v E V} together with
the coloring is an interval k-representation for G. We say Zj = {I(v) : v E Vi}
is an interval class of the representation and that both v and I(y) belong to
interval class i. We may consider the collection of intervals in a representation
as a family T {Zi,..., Zfc}, where denotes interval class i, for % = 1,2,..., k.
19


So, an IkG can be thought of as the intersection graph of a family of intervals of
M in k classes with vertices adjacent if and only if their intervals intersect and
belong to different classes.
Recall that a graph T is perfect if for every induced subgraph H of T,
x(H) (v(H), where x(G) is the fewest number of colors needed to properly
color G and ou(G) is the number of vertices in the largest induced clique of G.
Not every proper coloring of G is conducive to an interval ^-representation. For
example, if the vertices of a 4-cycle are each assigned to a different interval class,
then no interval ^-representation is possible. The reasoning is the same as that
forbidding a 4-cycle from being an interval graph. But because IkGs are perfect,
which we prove in this chapter, if G is an interval fc-graph, then in order to give
an interval /^-representation for G, we need only co(G) interval classes; more than
uj{G) interval classes will not allow for a representation when u(G) does not.
A probe interval graph (PIG) is a graph G = (V, E) in which V can be
partitioned into subsets P and N called probes and nonprobes, respectively, and
to each iigk there corresponds a unique interval I(v) Ct with uv G E if and
only if I{u) D I(v) ^ 0 and at least one of u, v belongs to P. For G = (V, E) a
PIG, the collection {I(v) : v G V} together with the vertex partition is called
a probe interval representation. Note that G(N) is an independent set, G(P)
is an interval graph, and any interval graph is a probe interval graph with
N = 0. These observations do not characterize probe interval graphs, but they
do characterize a related class of graphs. A graph G = (V, E) is an interval
split graph if V can be partitioned V = (Ui,U2) so that G(Ui) is an interval
graph and U2 is an independent set. Clearly, every PIG is an interval split
20


graph, but the converse is not true; see discussion regarding Figure 2.1. Also,
if a given partition V = (P, N) is forced, it may prohibit a graph from having a
probe interval representation, even though the graph may have a probe interval
representation using another partition. For example in any induced 4-cycle of
a PIG, exactly one of the pairs of non-adjacent vertices must be contained in
N. Hence, if in any induced 4-cycle, adjacent vertices are forced to belong to P,
then the graph is not a PIG. If G (V, E) is a PIG with partition V = (P, N)
forced, then we say that G is a partitioned, probe interval graph.
Proposition 2.1 If G is a probe interval graph, then in any induced 4-cycle
exactly one of the pairs of non-adjacent vertices must belong to N.
Proof: Let G = (V, E) be a PIG with C = {1, 2,3,4} C V and (1, 2,3,4,1)
an induced 4-cycle. Clearly no three vertices of C can belong to N and no ad-
jacent vertices can both belong to N. Assume that only one vertex belongs
to N\ relabel if necessary, and let 1,2,3 E P, and 4 6 JV. 1(1) n 1(2) ^ 0,
1(2) n J(3) 7^ 0 and 1(1), 1(3) must be disjoint. So, we may assume r(l) < 1(3).
But since 1(4) must intersect both 1(1) and 1(3), it intersects 1(2), forcing
4^2, contradiction. Thus, the only possibility is for 1,3 E N or 2,4 E N.
We assume 2,4 G iV and give a probe interval representation for G(C). Put
7(1) = [1,3], 7(3) = [4,6], and 7(2) = 7(4) = [2,5]. The result follows because
any induced subgraph of a PIG is a PIG.
Theorem 2.9 and Theorem 2.7 will show that a 4-cycle is the largest cycle a PIG
can have as an induced subgraph.
Now we put Proposition 2.1 to work and give an example of an IkG and an
21


e /

ht c


d
a
Figure 2.1: An IkG that is not a PIG
interval split graph that is not a PIG. M2 of Figure 2.1 is an IkG, and an interval
split graph, but not a PIG. Beside M2 is an interval ^-representation, so M2 is an
IkG. Put U\ = {6, c, e, /} and U2 = {a, d}; M2(Ui) is a path on 4 vertices, which
is an interval graph, U2 is an independent set, so M2 is an interval split graph.
We prove that M2 is not a PIG using Proposition 2.1. By way of contradiction,
assume that M2 is a PIG and consider {a, b, c, d}. By Proposition 2.1 either
a, d £ N or c, b G N. In the former case d, f G N is forced since {a, b, c, /}
induces a 4-cycle, but then adjacent vertices d and / are in IV; similarly, in
the latter case b and e are forced to be nonprobes, because {a, c, d, e} induces a
4-cycle. We have adjacent nonprobes in either case, contradicting the fact that
nonprobes form an independent set in a PIG.
2.2 A characterization for IkGs and a containment relationship
A set of distinct induced subgraphs Q = {Gx,..., Gt} of a graph G (V, E)
is consecutively ordered when for each v G V, if i < j < l and v G G{ D Gi, then
v G Gj. We will say that Q covers G if it forms an edge cover of G; i.e., if every
22


edge belongs to some H G Q. Theorem 2.2 supports Theorem 2.3 and both give
context to Theorem 2.6.
Theorem 2.2 (Fulkerson, Gross, 1965, [13]) A graph is an interval graph if and
only if its maximal cliques can be consecutively ordered.
A quasi clique Q in a PIG G = (P, N, E) is a set of vertices with all vertices
of Q fl P adjacent, and any vertex of Q D N adjacent to all vertices of Q D P.
A maximal quasi clique of G is a quasi clique that is not contained in any
larger quasi clique. A complete set of maximal quasi cliques of G is a collection
of maximal quasi cliques in which each maximal clique of G is in exactly one
maximal quasi clique of the set. A collection of sets is said to have the Helly
property if whenever a subcollection Si,...,Sk of them intersect pairwise, then
nil Si is nonempty. Any collection of intervals has the Helly property. Theorem
2.3 is purportedly a consequence of Theorem 2.2 and because intervals have the
Helly property, see [35], Aside from this claim, there is no published proof of
Theorem 2.3, so we attempted to give one here, but the result we proved turned
out to be a stronger result; it is Theorem 2.4.
Theorem 2.3 (Zhang, 1994, [48]) An interval split graph G = (f/i, I72, P), C/2
an independent set, is a probe interval graph with respect to the same partition
Ui = P, U2 = N if and only if there is a complete set of maximal quasi cliques
that can be consecutively ordered.
Theorem 2.4 Let G = (Pi,I72,P) be a graph with G(t/2) independent set.
G is a PIG with P = U\, N = P2 if and only if G has an edge cover of quasi
cliques that can be consecutively ordered.
23


Proof: Let G = (U\, U2, E) be a graph with G(U2) an independent set.
Suppose Q = {Qi,..., Qs} is an edge cover of quasi cliques that is consec-
utively ordered. For each v G V define an interval I(y) |7(u),r(u)], where
l(v) = min{i : v G Qi}, and r(v) = max{j : v G Qj}, and N = U2,P = U\.
We claim {I(v) : v G V} together with the partition is a probe interval repre-
sentation for G. By definition of Q, for vertices u, v, we have I(u) Pi I(v) 7^ 0 if
and only if they belong to the same quasi clique, in which case uv G E unless
u,v G N = U2] but if it, u G N, then the intersection of their intervals does
not induce an edge in G. Thus, G is a probe interval graph with U\ = P and
U2 = N.
Now suppose G has a probe interval representation I = {I(v)}vev, and let
ri < r2 << rrn he the distinct right endpoints among intervals of I. Define
Qi to be the subgraph of G induced on [Jp, -neiiv)} v:> this subgraph is a quasi
clique, and the collection Q = {Qi}t£l1 is a consecutively ordered collection of
quasi cliques that covers the edges of G.
We will use the next result regarding where maximal cliques of a PIG are
in a consecutively ordered cover of quasi cliques to prove a necessary condition
for bipartite PIGs in Chapter 4.
Lemma 2.5 If G is a PIG, then the consecutive cover of quasi cliques Q can be
made so that each maximal clique of G is contained in exactly one quasi clique
of Q.
Proof: Let G be a PIG with Q a consecutively ordered edge cover of quasi
24


cliques defined by the distinct right endpoints of the probe interval represen-
tation for G as in the proof of Theorem 2.4; that is, Qi = G (Up, -r-e/p)} 'u)>
where ri < < rm are the distinct right endpoints. A maximal clique C of G
is a quasi clique containing at most one nonprobe. No C is contained in more
than one Qi because each vertex v with interval [l{v), rf\ will not be contained
in Qi+i. Now we show that every C is contained in at least one Qi. Assume C
consists of P0 C P and n e N. We must have -pep01(p) ^ Hn) containing
some common point q by the Helly property. Either q = r* or q £ (ri; Ty+i), for
some i. If q = ri, then C is contained in Qi. If q e (r*, ri+1), then C is contained
in Qi+1, since r(v) > r*+1 for all v G C. We have shown that C is in at least one
and no more than one Qi, for some i.
The next result has a flavor similar to that of Theorem 2.4 and Theorem 2.2
and will serve in showing that any probe interval graph is an interval fc-graph.
Theorem 2.6 A graph is an interval k-graph if and only if there exists a cover
of complete r-partite subgraphs that can be consecutively ordered, where 1 < r <
k for each subgraph.
Proof: Suppose G = (V, E) is an interval fc-graph with representation
X = {I(v) = [£(,c),r(u)] : v G V}. Index the vertices vi,v2, ,vn so that
r(vi) < r(vj) whenever i < j. For each j e {1,... ,n}, the collection of vertices
whose intervals contain r(vi) induces a complete r-partite subgraph Gi of G, for
n
1 < r < fc. Define Q = (^J Gp, then Q is certainly a cover of G, and the indexing
i1
is a consecutive ordering since any collection of intervals has the Helly property.
25


Now suppose that Q = {G\,... ,Gm} is a cover of consecutively ordered
r-partite subgraphs. Assign the vertices of each partite set of G\ to a distinct
interval class and start by letting each interval representing a vertex of Gi be
a point at 1 6 1. For integers i = 2,..., to, and for each vertex in Gi-i D G
extend its interval from i 1 to i G R. Assign the vertices in each partite set
of Gi \ (Gi-i D G{) to their corresponding interval classes. This can be done
unambiguously because a vertex in Gi must be in the same interval class as any
non-adjacent vertex in Gi, and vertices belong to distinct interval classes when-
ever they are adjacent in Gi. Since each Gi e Q is an induced subgraph, the
interval class assignments will not change for vertices contained in two or more
consecutive subgraphs. Therefore, we have an interval ^-representation for G.u
Now we can easily show that the class of probe interval graphs is contained
in the class of interval /c-graphs.
Theorem 2.7 Every k-chromatic probe interval graph is an interval k-graph.
Proof: Suppose G = (P, N, E) is a /c-chromatic PIG; then G has a cover
of maximal quasi cliques Q that can be consecutively ordered. Each maximal
quasi clique Q E Q is a where t is the number of nonprobes in Q and
P n Q is a clique, in other words, a ATi,...,i- Thus, Q is a cover of complete
multipartite subgraphs that is consecutively ordered.
Figure 2.2 is an illustration of Theorem 2.6. The four distinct right endpoints
indicated by the vertical dotted lines induce, in the consecutive order, a Ad,2,
26


Figure 2.2: An IkG and its consecutive multipartite subgraphs
a a K2<2, and a iGil;2 on the sets {1,2,4}, {1,3,4,5}, {4,6,7,8}, and
{6,7,8, 9}, respectively.
Although every PIG is an interval split graph, and every PIG is an IkG, the
containment relationship between interval split graphs and IkGs is not known.
In the next section we prove IkGs are weakly chordal (and hence contain no
induced cycle of length greater than or equal to 5), but a cycle of any length
is an interval split graph. Thus, there are interval split graphs that are not
IkGs, but the converse is not known but has not received much attention. We
summarize the known containment relationships addressed here by:
interval graphs C PIGs C IkGs
with a 4-cycle and M2 serving as separating examples.
2.3 Some properties of IkGs
27


Recall that a graph G is perfect if x{H) = ^(H) for every induced subgraph
H of G. Let ik(G) be the fewest number of interval classes needed in an interval
/^-representation of G, and define ik(G) = oo in case G is not an interval k-graph.
The next result shows that ik(G) = co(G) provided that G has an interval k-
representation. Note: in any proper coloring, the vertices in a clique each require
a different color, hence, u(G) < x(G), for any graph G. Given G an IkG, the
next result shows how to color G with cu(G) colors, showing x(G) = w(G). Since
IkGs have the property that any induced subgraph of an IkG is an IkG, this
is called the hereditary property, IkGs are perfect. The result also shows that
the number of interval classes needed to give an interval /^-representation for a
graph if it is an interval k-graph is the size of the largest clique.
Theorem 2.8 Interval k-graphs are perfect.
Proof: Let G be an n-vertex IkG with interval ^-representation X = {I(v) :
v £ V(G)}, where the vertices have been colored with {1,2,..., r}, r pose I(vi) = [k,rj\, and relabel the vertices so that k < lj if i < j. Sequen-
tially color the vertices in order Vi,V2,... ,vn, recoloring if necessary, using the
first available color. We claim that this algorithm produces an w(G)-coloring.
Clearly v\ can get color 1. Assume that at step i we need color k, that is, Vi
needs color k, and that up to this point we have used no more than co(G) colors.
Since Vi requires color k, it must be adjacent to ..., tiik_1 colored with
1,2, ...,k 1, where ix < i2 < < ik-\ < i- Thus, r^ > k for each j satisfying
1 < 3 < k 1, and so G^v^, vi2,..., Uifc_15 Ui}) is a clique on k vertices, in other
words, lo(G) > k. By induction, this procedure will produce an w(G)-coloring.
Therefore, x(G) = w(G) for G an interval /c-graph.
28


A graph G is weakly chordal if neither G nor G contains an induced cycle of
length greater than 4. The next result shows IkGs are weakly chordal (and hence
so are PIGs) and provides us with examples of forbidden IkGs: the complement
of a 6-cycle in Figure 2.3, and any cycle of length 5 or greater.
Theorem 2.9 Interval k-graphs are weakly chordal.
Proof: Assume C = (t>i, v2,..., vn) is a cycle of length n > 5 in an IkG, G.
Assume that I(vi) is a leftmost interval in the representation for C and without
loss of generality that r(ui) = 1 and I(yi) £ X\. Then i(w2) < 1 and l(vn) < 1,
and since u2 w vn, they belong to the same interval class, say Z2. 7(u3) must
belong to an interval class distinct from Z2, and (I(v2) n 7(u3)) fi I(vn) = 0
necessarily, so r(v2) > l(v3) > r(vn) > 1. I(vn-i) must belong to a class distinct
from Z2, and 1 < l(vn_i) < r(vn) implies that l(vn-i) £ 7(u2), so C has a chord.
Suppose now that G contains Cn, the complement of a chordless cycle of
length n. If n = 5, then there is a 5-cycle in G, which contradicts what we have
just proved. For n = 6, we prove that the graph in Figure 2.3, which is the
complement of a 6-cycle is not an IkG. We use the labels in Figure 2.3 for the
proof. By symmetry, we may assume that 1(2) fi 7(4) lies between 7(1) fi 7(6)
and 7(3) H7(5). One of 7(2), 7(3) belongs to a class different from that to which
7(6) belongs; so no point in rij=i)2i37() can lie to the left of r(6). If 7(5) fl 7(6)
extends to the left of rij=ii2)37(i), then 5 -B- 1 or 5 £> 2, while if 7(5) Pi 7(6)
extends to the right of rh=li2j37(i), then 6 £) 2 or 6 -H- 3.
We make the observation that there are only two distinct ways to decompose
29


2
4
Figure 2.3: The complement of a 6-cycle, which is not an IkG
a 4-cycle into consecutively orderable complete multipartite subgraphs: as a K2>2
or as two copies of Ki>2. In either case, at least one pair of nonadjacent vertices
must belong to the same interval class, and these intervals must intersect.
For n > 6, consider chordless 4-cycles in Cn = (1,2,..., n). We must make
interval class assignments for the vertices of Cn based on their adjacencies. We
may assume for the cycle (1,4,2, 5) that vertices 1 and 2 belong to the same
interval class. The cycle (2,5,3,6) must contain a nonadjacent pair of vertices
whose intervals belong to the same class. This pair cannot be 2,3 since 3 and 1
are adjacent in Cn, so 5 and 6 belong to the same class. In the cycle (2,6,3,7,2),
6 and 7 cannot belong to the same interval class, since 7 <- 5. Therefore Cn
does not have an interval ^-representation.
In [23] Hayward showed that weakly chordal graphs are perfect and so we
have another proof of Theorem 2.8.
2.4 Forbidden substructures
An asteroidal triple of a graph G is a set of three vertices with the property
that between any two there is a path in G that avoids the neighborhood of the
third vertex. In [4] it is proved that a chordal graph is an interval graph if and
30


Figure 2.4: Examples of ATEs
only if it does not contain an asteroidal triple in any induced subgraph. There
is a forbidden substructure for IkGs that is, in principle, similar to an asteroidal
triple. Define the neighborhood of an edge to be the union of the neighborhoods
of its vertices. An asteroidal triple of edges (ATE) in G is a set of three edges
such that for any two there exists a path in G that avoids the neighborhood of
the third edge. Figure 2.4 has two bipartite graphs each with an ATE.
Theorem 2.10 If a graph G has an asteroidal triple of edges, then G is not an
interval k-graph.
Proof: Let G = (V,E) be an IkG and X\X2, yiV2, ziz2 £ E an ATE of
G. We use the notation I(uv) = I(u) n I(v), where u,v G V and uv G E. If
I(xix2) fl I{yiy2) 7^ 0, then each of N(yf), N(y2) contains at least one of xi,x2
and no path from z\z2 to X\X2 can avoid the neighborhood of y\y2- Therefore,
I(xix2), I(yiy2), and I(ziz2) are disjoint, and we may assume that I(yiy2) lies
between the other two. Let P = (pi,... ,pm) be a path between xix2 and ziz2.
Then /(yip2) D (I{pi) Ul(p2) U Ul(pm)) C I(pi) for somep^ G P. At least one
of I(yi),I(y2) belongs to an interval class different from that containing I(pi),
31


so pi E N(yi) U N(y2), contrary to assumption.

This gives an interesting corollary regarding PIGs.
Corollary 2.11 Probe interval graphs are ATE-free.
The converse of Theorem 2.10 is not true unless k = 2 and we consider only
trees, see Chapter 3. In the proof of Theorem 2.9 it is shown that the ATE-free
graph in Figure 2.3 is not an IkG. We present two graphs that are minimal
forbidden subgraphs for IkGs, one with an ATE and the other ATE-free. Refer
to Figure 2.5. H2 is ATE-free and we use Theorem 2.6 to show that it is not an
IkG, while the thick edges of Hi are an ATE.
Figure 2.5: Two minimal forbidden graphs for IkGs
Consider H2 5; we may take the subgraphs induced on {1,3,4}, {2,3,4},
{3,4,6,8}, {3,6,8,10}, {7,8,10}, {8,9,10} in that order as our consecutive
complete multipartite subgraphs. For H2, notice that the clique on {3,6,7}
must be contained in both 4-cliques, which necessitates taking the 4-cliques
consecutively. Three of the 5 pendant vertices must be taken in subgraphs either
32


before or after the 4-cliques, and it is evident that 3 subgraphs containing a triple
of pendants cannot be ordered without taking a 4-clique within that ordering.
Both graphs of Figure 2.5 belong to families of minimal forbidden subgraphs for
IkGs. A forbidden subgraph characterization for interval fc-graphs seems rather
difficult at this time. However, we posit the following conjecture. We require
that G have x(G) = 3 because in Chapter 3, we will see that the following is
true: For G bipartite and chordal, G is an interval bigraph if and only if it has
no ATE. Also H2 of Figure 2.5 is 4-chromatic, chordal and has no ATE.
Conjecture 2.12 Let G be a chordal graph with x(G) = 3 and no ATE. Then
G is an interval 3-graph.
2.5 Applications of IkGs
Let C be a set of chemicals or compounds, each with temperature con-
straints for its storage. Suppose subsets of the chemicals react with one another
or pose undesired contamination threats when combined, and therefore cannot
be stored together. Break C into {Ci,..., Ck}, where each Cj consists of chem-
icals or compounds that cannot be stored together regardless of temperature
compatibility. Furthermore let C = {%}, where the first subscript indicates to
which Ci the compound or chemical c^- belongs and the second subscript enu-
merates the members of Ci, so 1 < j < \Ci\. The temperature constraint for
each Cij will be a closed interval of K: ) = [l(ij),u(ij)]. Model the storage
of C by the graph G = (C, E), where CijCrs E if and only if i(cy) fl t(crs) ^ 0
and i ^ r. In other words, G is a graph with C as its vertices and vertices are
adjacent if they may be stored at the same temperature and they pose no threat
33


of reaction or contamination when stored together. Observe that a clique in
G corresponds to a group of chemicals that can be stored together and at the
same temperature. Therefore, the minimum number of temperature regulated
containers required to store C is precisely the minimum number of cliques which
cover the vertices of G.
Now consider interval 2-graphs; i.e., interval bigraphs. A typical application
of a bipartite graph G = (W, J, E) is to have one partite set W = {w} correspond
to workers and the other J = {j} to jobs with wj £ A1 if w can do j. We modify
this application to include constraints on when the workers work and when the
jobs can be done. We assume that the ability of a worker to do a particular job
rests on availability only; that is, if worker w is employed while job j can be
done, then w can do j. Here we assign intervals to each vertex of G = (W, J, E):
I(v) = [b(v),e(v)], corresponding to the times the workers shift begins and
ends or when the jobs availability begins and ends. So, wj E E if and only if
I(w) fi I(j) / 0, and an optimal assignment of workers to jobs corresponds to a
maximum matching in G\ as in the standard application.
34


3. Interval Bigraphs
Our attention in this chapter is restricted to interval 2-graphs, or interval
bigraphs. We give a characterization of them via their reduced adjacency matri-
ces, present the characterizations in the current literature, and summarize the
progress on the forbidden induced subgraph characterization.
3.1 Background and definitions
An interval bigraph (IBG) is the intersection graph of two distinct families
of intervals with vertices adjacent if and only if their corresponding intervals
overlap, and the intervals belong to different families. In other words, interval
bigraphs are interval 2-graphs. The two families of intervals that represent an
IBG will be called the interval bigraph representation, and with G = (V., E) an
IBG, I(x), x E V denotes the interval corresponding to vertex x, while l{x) and
r(x) denote the left and right endpoints of I(x), respectively. Figure 3.1 shows
that H10 is an IBG, since to the right of it is an interval bigraph representation.
Figure 3.1: H10 and an interval bigraph representation.
Recall that a probe interval graph can be defined as follows. A graph G =
(V, E) is a probe interval graph if there is a partition of V = (P, N) and an
35


interval can be assigned to each vertex so that vertices are adjacent if and only
if their corresponding intervals overlap and at least one of the vertices belongs to
P. So, if at at least is replaced by exactly, then we have another definition for
IBGs. This natural extension of the notion of a probe interval graph is what led
us to study IBGs. Because of the application probe interval graphs model, we
originally called IBGs probe-clone bipartite interval graphs. But this unwieldy
name bogged us down and so they became interval bigraphs. This nomenclature
allowed us to find pre-existing research by Harary, Kabell and McMorris, [22],
and H. Muller, [37], which makes our research more relevant.
In this chapter, we will characterize IBGs by the structure of their reduced
adjacency matrices, and start a thread that we continue for the rest of this
thesis: we give matrix characterizations for interval point bigraphs and unit
interval bigraphs in the chapters that follow. As a corollary to Theorem 2.6
IBGs are characterized by the existence of a consecutively orderable edge cover of
bicliques. Recent work by P. Hell and J. Huang, [24], identifies the complements
of IBGs with a class of restricted 2-clique circular arc graphs.
The (0, l)-matrix property that will turn out to describe precisely the struc-
ture of the reduced adjacency matrix of an IBG is as follows. This property
was introduced in [10] to characterize interval digraphs. We use properties of
bipartite graphs to get Theorem 3.4. A matrix with a zero-partition (ZP) is a
(0, l)-matrix with independent row and column permutations after which each
zero can be labeled R or C so that every entry below a C is a C and every entry
to the right of an R is an R. Figure 3.2 shows an example in which rows 6 and
7 are swapped and then columns 6 and 7 are placed between columns 1 and 2
36


1 0 0 0 0 1 0 1 1 1 1
1 1 0 0 0 1 1 11 11 1 1 1 1
0 1 1 0 0 1 1 11 11, 1 1 1 1
0 0 1 1 0 1 i c _ 11 I [ 1 1 1 1
0 0 0 1 1 1 1 1 1 1 I 1 ]. 1 1
0 0 0 0 1 0 1 1 1 1 1
0 0 0 0 0 1 1 1 1 1 1
1 1 R R R R R
1 1 1 1 R R R
c 1 1 1 1 R R
c 1 1 c 1 1 R
c 1 1 c c 1 1
c 1 1 c c c R
c c 1 c c c 1
Figure 3.2: A zero partitionable matrix
to obtain a zero partition. We will show that IBGs are precisely those bipartite
graphs whose reduced adjacency matrices are zero-partitionable.
A more restricted property for (0, l)-matrices is the consecutive ls property.
A (0, l)-matrix has the consecutive 1 s property for rows if the columns can be
permuted so that the ls in every row appear consecutively; the consecutive ls
property for columns is defined similarly. Tucker investigated the structure of
(0, l)-matrices with the consecutive ls property for columns by determining
which bipartite graphs do not have the consecutive ls property for columns in
their reduced adjacency matrices. He dubbed his result a structure theorem
for the consecutive ls property, see [45]. A bipartite graph G = (X,Y,E)
is X-consecutive if the vertices in X can be ordered so that for each y E Y,
N(y) is a consecutive set in X with respect to the ordering found for X. Note:
37


in this thesis rows of the reduced adjacency matrix are always indexed with
X, and columns with Y, when representing a bipartite graph with a matrix;
conversely, rows will correspond to X when representing the matrix as a graph.
Clearly, with this convention, an X-consecutive bipartite graph has a reduced
adjacency matrix with the consecutive ls property for columns. The following
result of Tuckers will be used in this chapter and the next. It shows that
a structural characterization for the consecutive ls property for columns is
tantamount to determining all bipartite graphs G = (X, Y, E) with an asteroidal
triple contained in X, where the bipartition is predetermined.
Theorem 3.1 (Tucker, 1972, [45]) A bipartite graph G = (X,Y,E) is X-
consecutive if and only if there is no asteroidal triple of G contained in X;
i.e., A(G) has the consecutive ls property for columns if and only if G(A) has
no asteroidal triple contained in X.
We will use Theorem 3.1 to show that having a zero partition is a property less
restrictive than having the consecutive ls property for rows or for columns. To
this end we present the following proposition which shows that the class of zero
partitionable matrices properly contains those with consecutive ls in rows or in
columns. Note that we use Theorem 3.1 to show that the consecutive property
for columns is not present by reversing the roles of X and Y.
Proposition 3.2 If a (0,1)-matrix has the consecutive 1 s property for rows or
for columns, then it has a zero partition, but not conversely.
Proof: Let Mbeanmxn- (0, l)-matrixwith the consecutive ls property
for rows (columns). Let a* {fif) be the first column (row) in which a 1 appears
38


in row i (column j), 1 < % < m (1 < j < n). Permute rows (columns) so that
{;} (fij) forms a nondecreasing sequence. Now, label all zeros to the right of
(below) the last 1 in each row (column) with an R (a C). The matrix that
results exhibits a zero partition.
The matrix
T 0 0 0 0 ' 1 R R RR
11100 1 1 1 RR
01000 has a zero partition: G 1 R RR
00110 C C 1 1 R
00101 C c 1 C 1
but it is the re-
duced adjacency matrix of H10 in Figure 3.1, which has an asteroidal triple on
each of the sets {a,i, e}, {b,j, /} and each one is contained in a distinct partite
set of H10. Therefore, by Theorem 3.1, A(.010) separates the classes of zero
partitionable matrices from that of (0, l)-matrices with consecutive ls in rows
or in columns.

3.2 Characterizations for IBGs
First we completely characterize the trees that are IBGs. Note that IBGs
have the hereditary property, that is, any induced subgraph of an IBG is an IBG,
so if an induced subgraph of G is not an IBG, then G is not an IBG. We use
(no, vi, v2,.. vn) to denote a path of length n, and T(H) denotes the interval
representation for the subgraph H. For v a vertex, the size of v's neighborhood
N(v) is |IV(u)| = deg(u).
Theorem 3.3 A cycle-free graph G is an IBG if and only if it has no NL10 of
Figure 3.3 as an induced subgraph.
Proof: First, we show that NL10 is not an IBG. Suppose, on the contrary
that it is and construct a longest path. Without loss of generality, and using
the labeling in Figure 3.3, suppose we have the interval bigraph representation
39


NL10 3 T
i 9
h o
o6o
a b c d e f g
Figure 3.3: NL10: The smallest tree that is not an IBG and the only forbidden
subgraph for cycle-free IBGs
for P = (a,b,c,d,e, /). Because i is not adjacent to c or e, and the length
of P is sufficiently large, I{i) C 1(d). Hence I(j) is forced to intersect 1(d),
contradiction.
Now, suppose G is a cycle-free graph with NL10 not an induced subgraph.
Let P = (x0, Xi,..., xn be a path of maximum length. Since P has maximum
length deg(rr0) = 1 = deg(:cn), and deg(it) = 1, for any u E (N(xi)\)P U
(N(xn-1) \ P). Since G has no induced NL10 deg(u) = 1, for any vertex v
at distance 2 from any vertex of P. Create T(P) with I(xi) = [i,i + 1], for
i = 0,... ,n and place deg(xi) 2 equidistantly spaced points in (1, 2) for its
neighbors, and similarly deg(a;n_i) 2 points in (n 1, n) for £n_is neighbors.
Now for each X{, 2 < i < n 2, with deg(xj) > 2, put deg(£j) 2 equidistantly
spaced points in (i,i + 1) for X{s neighbors at distance 1 from P, and for any
vertex at distance 2 from Xi not on P, make its interval a point equal to its
(unique) neighbors point. G is cycle-free, and hence bipartite, so the natural
bipartition of G defines the interval classes and the construction renders G an
IBG.
Now for the matrix characterization for IBGs in general. A standard zero-
partition is a zero-partition in which an ambiguous 0 is labeled C if it lies on or
40


below the last full diagonal, while an ambiguous 0 above this diagonal is labeled
R. In symbols, entry (i,j) e D' = {(1, n (m-1)), (2, n (ra-2)),..., (m,n)}
or below D' is labeled C if it can be labeled either R or C, and labeled R if it lies
above D' and can be labeled either R or C. We make the following assumptions
at the outset: a (0, l)-matrix has m rows and n columns, m < n (since the
class of zero-partitionable matrices is closed under taking the transpose), any
zero-partitionable matrix will be in standard form. The following example shows
how assuming that a zero partition be in standard form precludes ambiguity,
since some of the entries in the lower right can be labeled R or C. A standard
zero-partition for the matrix A is given below:
Till 1111' "11111111'
10000000 1 CCCCRRR
10000110 1CCCC1 1 R
10000100 ICCCC1CR
10000000 1CCCCCCC
Theorem 3.4 G is an interval bigraph if and only if its reduced adjacency ma-
trix has a zero-partition.
Proof: (=>) Let G = (X, Y, E) be an interval bigraph with X =
{Ix, Iy}X£X,y£Y as an interval representation with Ix = \l(xi),r{xi)\ for i =
1,..., \X\ = m, and Iy = [l{yj),r{yj)] for j = 1,..., \Y\ .= n. Index X UY
so that l{xi) < l(x2) < < l(xm) and l(yf) < < l(yn)- Form A(G), the
reduced adjacency matrix of G with row i corresponding to X{ and column j
corresponding to yj. This ordering of rows and columns will enable us to label
the zeros of A(G) in accord with a zero-partition.
Assume entry (i,j) 0. Then either r(xf) < l{yf) or r{yf) < l(xi) but not
both since l(xi) < r(xf) and l(yf) < r{yf). If rixf) < l(yj), then label entry (i,j)
41


with R. Every entry to the right will be a zero which we may label R. To see
this suppose (i, j + s) = 1, for s > 1. Then r{xi) > l(yj+s), but l(yj+s) > l(yj)
and this implies r(xi) > l{yj)m, contradiction. If (i,j) = 0 and rijjj) < l(xi), then
label entry (i,j) with a C. Now suppose (i + s,j) = 1, again with s > 1, then
r{Vj) > Kxi+s) and since l(xi+s) > l(xi) we have r(yj) > l(xi); contradiction.
So every entry below (i, j) is a zero we may label C. This shows that A(G) has
a zero-partition.
(4=) We create an interval representation for the bipartite graph, G =
(.X,Y,E), obtained from A = a zero-partitionable matrix in standard
form, in which xy G E if and only if aitj = 1. We assume m = \X\ < \Y\ = n,
and that there are no all-zero rows or columns, since the intervals for isolated
vertices may always be placed with no trouble. All intervals will be open subsets
of the segment (|, m + 1). Open intervals will serve to preclude singleton point
intersections. The standard partition serves to relieve problems that would oth-
erwise arise from sparseness in the lower right of the matrix, and of course to
eliminate ambiguity in the labeling. Essentially, {Iy}yeY is created so that Ix
may be inserted so that Ixf\Iy = 0 whenever (i, j) = 0, and Ixr\Iy 7^ 0 whenever
(*, j) = 1. We define the following symbols which allow us to use the information
implicit in the zero-partition:
first column in which a C appears in row %
0 if there is no C in row i
' last column in which a C appears in row i
0 if there is no C in row i
first column in which an R appears in row i
n + 1 if no R appears in row i
We define the analogous indicators for columns:
42


, f the last row in which a R appears in column j
Pi \ 0 if there is no R in column j
/ / the first row in which a C appears in column j
^i \ m + 1 if there is no C in column j
First create {Iyi}i=i by specifying left and right endpoints of the interval Iyj cor-
responding to yf Iyj = (l(yj),r(yj)). Define Sj = |{(i, j) : aLj = R, 1 < i < p'}|;
note that Sj-i = Sj when N(yj) and N(yj-1) agree on xi, x2,..., xpi..
Initialize r(y0) = 1, and l(yn+i) = m+ 1.
For j e {1,..., n};
If P'i = P'i-1;
Put l{Vj) = 4 + \ + r{vj) = 7';
Else put l(yj) = p'- + r(%) = 7'-.
Now we create {1^}^, by specifying left and right endpoints for each in-
terval IXi corresponding to :Cj.
For i e {1,... ,m};
Put l(xi) = max{r(yj) : fa < j < 7t,ay = C}, r(xj) = min{Z(?/j) : Pi < j <
n + 1}.
Claim 1 The sequence {s^} is nondecreasing.
Choose an arbitrary row i and column j, and suppose Sj+i < Sj. Then above
row i we have at least one or fewer f?s in column j + 1 as we have in column
j. This implies we have a submatrix of the form
Therefore Sj < Sj+1.
Claim 2 For each i, l(xi) < i, and r(xi) >i + \.
First we show l(xi) < i. If fa = 7i = 0, then l(xi) = r(r/o) = 1 < i
R *
RR
where G {1, C}.
43


for all i. Otherwise we have 1 < fa < 7* < n. Recall that r(yf) = 7'-, and
since, by assumption, there is a C in row i, 7'- < i. Hence, for each j satisfying
fa < j < 7i, and a^j = C, r(yj) < i. Therefore, l(xi) < i.
Now we show that r(xi) >i + \. If pi = n + 1, then l(yn+1) = m + 1 > i;
so r(xi) > i + Otherwise we have 1 < Pi < j < n, and for each j, l(yj) >
p'- + | > i + \ since a^j = R when j satisfies Pi < j < n.
Claim 3 \Iy.\ > ^77 for all j, and |Ix.\ >
By design, \Iyj \ > ^7, and \IXi\ > \ by Claim 2.
Claim 4 The sequence {p' | is nondecreasing.
Assume j < k and p'- > p'fc. Then we have a submatrix of the form
* R
R 1
or
* R
RC
* C
C R
or
. Therefore, p'- < p'k whenever j < k.
Claim 5 The sequence {7^} is nondecreasing.
Assume i < j and 7{ > 7j. Then we have a submatrix of the form
" cl
g 1 So 7^ < 7j whenever i < j.
Claim 6 The value mm{l(yj) : pi < j < n+1} = l{yPi), a,nd hencer(xi) = l{yPi)-
If pi = n +1, then r(xi) = l(yn+1) = m +1. Fix i arbitrarily and choose k so
that pi l(yk) since {s^} is nondecreasing by Claim 1. If p'p. 7^ p'k, then p'p. < p'k by Claim
4, and so l(yPi) =p'p. + \< p'. + 7(yk).
We will now prove that the algorithm is correct by verifying IXi n Iyj 7^ 0 if
and only if a^j = 1. We consider several cases each for [a^j = 0 => IXir\Iyj 0]
and [(i,i) = 1 => IXi O Iyj 7^ 0], which span Claim 7 and Claim 8.
Claim 7 If ai j = 0, then IXi D Iy, = 0.
We deal with = C and ay = R separately.
44


Case 1 Assume ay = 0 and is labeled C. Then r(z/j) < z and l(yj) < % 2r^\
But l(xi) > z since r(yj) < % and so l(yj) < r(yj) < l(xi) < r(xi). Therefore,
Ix. Pi Iyj = 0, since all intervals are open.
Case 2 Assume ay = 0 and is labeled R. Then r(xj) = min{7(yfc) : fa < k <
n + 1} = l(yPi) < l(yj) by Claim 6, since Pi < j < n + 1.
Claim 8 If ay = 1, then IXi fl Iyj "7 0-
We break this into the cases (1) j < fa, (2) i < j < 7j, and (3) 7i < j <
Pi < n + 1.
Case 1 (j < (f>i and ay = 1) By definition 7'- > i; and p'- < i otherwise we would
have [= [Si. So | 2 -1- 1 2 1 2m+l
Lvrjj L~J
while /(zj) < i and r(xi) > i + \ by Claim 2. Therefore, r(xi) > l{yf) and
l{xi) < r(yf), which implies IXi C Iy. 0.
Case 2 ( z; and p'- < z otherwise
as in Case 1. So l{yf) < i, r{yy) > i + 1, < i, and
r(xi) > i + Therefore, IXi fl Iyj 0.
Case 3 (ay = 1 and 7* < j < Pi < n + 1) As in Case 1 and in Case 2, 7'- > z;
but pt < i is not forced. However, if p'- < z, then we may argue as in Case 2.
Hence we will assume p'- > z, which gives the following set-up:
(i,0i) CC
(PjAi) {p'i,li) * R
j k
i C--- 1 R
pfj R R
where pi < k < n + 1. Let p'- = i + p for p > 1. The algorithm gives
% + p + | < Z(z/j) < z + p + | + . Since Z(zj) < z from Claim 2, we have
l(xi) < l(yj). Assume p'- 7^ p'_x and /(?/.,) = z+ p+ | = p'+|. The algorithm
45


puts r(xi) = min{l{yk) Pi < k < n + 1}, and by Claim 6, min{l(yk) Pi < k <
n + 1} = l{yPi). Let k = p{. If p'k = p'-, then we have l(yk) = p'k + \ + >
*+P+f+ 2m+l > P'j + 2= l(Vi) and n ^ If Pk P'j then Pfc > ^ ^
Claim 4, and so i(t/fc) = p'k + \ > p'j + l + \ > £(%) Again we have r(a;i) > /(%),
and /Xi n Iyj ^ 0.
Now we assume p'_x = pj, and so l(yj) = i + p + | + 2m+i If Pi = n + 1>
then r(xi) = £(yPi) = %n+i) = m + 1 > /(%). So /Xi n ^ 0. Assume p; < n.
We have the following set-up:
3 Pi n
i C-- 1 R R
p'i R-- R-- R
p'pi * R R
If p'p. j- p'j, then p'p. > p'-, and so rfa) = l(yPi) > i +p + 1 + f (%) If Pp* = Pj>
then l(yPi) =i+p+\+ ^ >i+p+\+ > i + p + \ + = %)
So either way we have r(xj > l(yj), which together with l(xi) < l{yj) (shown
above), gives IXi n Iyj / 0. This completes the proof.
Figure 3.4 is the interval representation created by the algorithm in the
above proof applied to the matrix A presented before Theorem 3.4 The intervals
for X are black and X{ (yj) corresponds to row i (column j).
46


^2 1 2 3 4 5 6

Figure 3.4: Interval bigraph representation for the matrix A
As another example, given the matrix
'1 0000 o' '1 R R RR R~
1 1110 0 1 1 1 1 R R
1 1101 0 1> 1 1 1 C 1 R
1 1100 1 1 1 1 CC 1
0 1000 0 c 1 c CC C
we get the following interval representation and graph in Figure 3.5. The graph
is taken from a set of forbidden subgraphs determined by Muller in [37] called
insects which we discuss below. An insect would have a vertex pendant to vertex
3 of Y.
We will now record the results that, at the time of writing this thesis, char-
acterize interval bigraphs. Chronologically speaking, Corollary 3.5 preceded
Theorem 2.6, see [6], and was then generalized to interval k-graphs, see [7], but
here we record it as a corollary. Recall that a set of distinct induced subgraphs
Q = {Gi,..., Gt} of a graph G = (V, E) is consecutively ordered when for any
v G V, if % < j < l and v G Gi D Gt, then v G Gj. We say that Q covers G if it
forms an edge cover of G.
47


2
3
4
5
6
Figure 3.5: An almost-insects interval bigraph representation
Corollary 3.5 A bipartite graph G is an IBG if and only if there exists a cover
for G consisting of bicliques that can be consecutively ordered.
Proof: An IBG is an interval 2-graph. Thus, by Theorem 2.6, there is a
cover of complete 1- or 2-partite subgraphs that can be consecutively ordered.
And conversely, if there is such a cover, G is an interval 2-graph, also by Theorem
2.6.
A circular arc graph is the intersection graph of arcs of a circle, see Figure 3.6
for an example. The clique cover number of a graph G is the minimum size of
a collection of induced complete subgraphs that contain all vertices of G. For
example, the complement of any connected nontrivial bipartite graph has clique
cover number 2. For short, we will say that G is a ^-clique graph and mean that
it has clique cover number k.
We have the following list of equivalences for bipartite graphs. The equivalences
(a), (b), (e), and (f) were found by Hell and Huang, see [24]. Das et al. in [10]
characterized interval digraphs with (d).
Theorem 3.6 The following statements are equivalent for a bipartite graph G =
48


5
Figure 3.6: A circular arc graph.
Va vb vc
Figure 3.7: Forbidden configuration for Theorem 3.6 (e).
(X, Y, E) with n vertices:
(a) G is an interval bigraph;
(b) G is a 2-clique circular arc graph in which no two arcs cover the whole circle;
(c) There is a cover ofG consisting of bicliques that can be consecutively ordered;
(d) The reduced adjacency matrix of G has a zero-partition;
(e) The vertices of G can be ordered Vi < V2 << vn so that there do not
exist a < b < c with va,Vb in the same partite set and vavc G E, but vbvc £ E
(see Figure 3.7);
(f) The vertices of G can be ordered v^ < V2 << vn so that there do not
exist a < b < c < d with the any of the four structures in Figure 3.8.
Now we prove Theorem 3.3 using the results we have developed. Assume
G is a cycle-free graph. If G has NL10 as an induced subgraph, then G has an
49


Va vb vc vd va vb vc vd va vb vc vd va vb vc vd
12 3 4
Figure 3.8: Forbidden configurations for Theorem 3.6 (f).
ATE and hence is not an IBG, by Theorem 2.10. Now assume that G has no
induced NL10. Then there is a path P from which all vertices have distance at
most two. Any vertex at distance one or two from P and not on P belongs to a
biclique isomorphic to Ki>2 or Kip. Beginning at one end of P take the edges of
P one at a time as induced bicliques, and when a vertex of degree greater than
or equal to three is encountered, take all of the Kip's and Kips incident with
that vertex in any order and then continue along P in the same fashion. This
creates a biclique cover that is consecutively ordered. Therefore, by Theorem
3.6 G is an IBG.
3.3 Forbidden Substructures and Subgraphs
According to the literature, cycles of even length greater than 4, the graphs
of Figure 3.10 and their generalizations, the three graphs in Figure 3.11, the
insects of Figure 3.12, and the bugs of Figure 3.13 are all of the bipartite graphs
that are forbidden subgraphs for IBGs that authors are willing to give explicitly.
We present these graphs in turn, and explain why or give references in which it
is explained why they are forbidden. We reiterate that this exposition will not
describe all forbidden subgraphs and substructures for IBGs since such a list is
not known.
50


Define the neighborhood of an edge e of a graph G to be N(e) = Uey(e) N(v).
Recall that an asteroidal triple of edges (ATE) of G is a set of three edges with a
path between any two that avoids the neighborhood of the third edge. In Figure
2.5 there are two bipartite graphs each with an ATE, and in Figure 3.10 the
graphs Di,Wi, and Tj, for i = 1, 2,3, have ATEs induced by the thicker edges.
Theorem 2.10 gives the following corollary, which was also proved in [37].
Corollary 3.7 If G is an IBG, then there is no asteroidal triple of edges in any
induced subgraph.
It turns out that the principle behind ATEs can be generalized to generate
many forbidden subgraphs for IBGs. Such a structure will be defined later and
can be found in graphs obtained from some of the finite families of finite sets in
Figure 3.9. Note that there are six infinite collections Ci,T{,Wi, Di, Mi, Ni, and
three separate families, r1; r2, T3 in Figure 3.9. Obtain a bipartite graph from a
family T in Figure 3.9 as follows. Let E = {Ai : 1 < i < k} consist of subsets of
{1,2,..., n}, n G Z+. Represent any T by a bipartite graph G = (X, Y, E) with
X = {xi,x2,, xn}, Y = {yi, y2,..., yk}, and XiVj e E if and only if i See Figures 3.10 and 3.11 for representations of the families as bipartite graphs.
Note that the representation of any Ci is simply a cycle of even length greater
than or equal to 6; C\ is a 6-cycle, C2 is an 8-cycle, and so on. The relevance
of this list will become clear once the following result of Trotter and Moore is
considered along with Theorem 3.6 (b.). Also, we record the list in Figure 3.9
here because it is useful for later reference; the order in which the sets are given
is, in a sense, as close to being conducive to an interval bigraph representation
or zero partitionable matrix as possible.
51


Theorem 3.8 (Trotter, Moore, 1979 [36]) A graph G is a 2-clique circular arc
graph if and only if G contains none of the bipartite graphs derived from the
finite families of finite sets in Figure 3.9.
So, by Theorem 3.6 (b.) IBGs are the complements of restricted 2-clique
circular arc graphs, and hence every graph obtained from a family in Figure 3.9
is not an IBG by Theorem 3.8. It is easy to see that any of the graphs of Figure
3.10 is minimal as a forbidden subgraph: simply remove a vertex and obtain
an interval bigraph representation. We illustrate with an example, but instead
of constructing an interval representation, we will take a set from Figure 3.9,
remove a set and show that its incidence matrix has a zero partition. Realize
that this is equivalent to exhibiting a zero partition for the reduced adjacency
matrix for the corresponding bipartite graph. Consider N3 of Figure 3.9. Re-
move the the set {10} giving N3 = {{1,2,3,4,5, 6,7}, {1,2,3,4, 5}, {1, 2,3}, {1},
{1,2,3,4,5, 6,8,10}, {1,2,3,4, 6,8}, {1, 2,4,6}, {2,4}, {2,9}}; now make an in-
cidence matrix with row i corresponding to element i and column j correspond-
ing to set j with respect to the order in which they are given. We get the
following matrix A(N3) which exhibits a zero partition after swapping rows 9
and 10:
52


Cl = {{1,2},{2,3},{3,1}}
C2 = {{1,2}, {2,3}, {3,4}, {4,1}}
C3 = {{1,2}, {2,3}, {3,4}, {4,5}, {5,1}}
= {{1,2}, {2,3}, {3,4}, {2, 3,5}, {5
T2 = {{1,2},{2,3},{3,4},{4,5},{2,3,4,6},{6}}
T3 = {{1,2},{2,3},{3,4},{4, 5},{5,6},{2,3,4,5,7}, {7}}
Wi = {{1,2}, {2,3}, {1,2,4}, {2,3,4}, {4}}
W2 = {{1,2},{2,3},{3,4},{1,2,3,5},{2,3,4,5},{5}}
Ws = {{1,2}, {2,3}, {3,4}, {4,5}, {1,2,3,4,6}, {2,3,4,5,6}, {6}}
Di = {{1,2,5}, {2,3,5}, {3}, {4,5}, {2,3,4,5}}
D2 = {{1,2,6}, {2,3,6}, {3,4,6}, {4}, {5,6}, {2,3,4,5,6}}
D3 = {{1,2,7}, {2,3,7}, {3,4,7}, {4,5,7}, {5}, {6,7}, {2,3,4,5,6,7}}
Mi = { 1,2,3,4,5}, {1,2,3}, {1}, (1,2,4,6}, {2,4}, {2,5}}
M2 = {{1,2,3,4,5,6,7}, (1,2,3,4,5}, {1,2,3}, {1}, {1,2,3,4,6,8},
{1,2,4,6}, {2,4}, {2,7}}
M3 = {{1,2,3,4,5,6,7,8,9}, {1,2,3,4,5,6,7}, {1,2,3,4,5}, {1,2,3}, {1},
{1,2,3,4,6,8,10}, {1,2,3,4,6,8}, {1,2,4,6}, {2,4}, {2,9}}
IVi = {{1,2,3}, {1}, {1,2,4,6}, {2,4}, {2,5}, {6}}
N2 = {{1,2,3,4,5}, {1,2,3}, {1}, {1,2,3,4,6,8}, {1,2,4,6}, {2,4}, {2,7}, {8}}
N3 = {{1,2,3,4,5,6,7}, {1,2,3,4,5}, {1,2,3}, {1}, {1,2,3,4,5,6,8,10}
{1,2,3,4,6,8}, {1,2,4,6}, {2,4}, {2,9}, {10}}
rx = {{1,3,5},{1,2}, {3,4}, {5,6}}
r2 = {{1},{1,2,3,4}, {2,4,5}, {2,3,6}}
r3 = {{1,2},{3,4},{5},{1,2,3},{1,3,5}}
Figure 3.9: The finite families of finite sets that represent the forbidden com-
plements of 2-clique circular arc graphs.
53


Figure 3.10: Some of the corresponding graphs of the families in Figure 3.9.
The thicker edges are ATEs or edge asteroids. The black vertices represent
elements and the white vertices represent subsets.
54


o 12
T 6
O 56
o 5
--o----6-----o
4 34 3 135 1 12 2

r2
2*
123 6------1> 1
3
O 135
u 5
i 5
r3
Figure 3.11: The graphs corresponding to ri,r2,r3
'111111100' '1111111 RR
111011111 111(711111
111011000 men rrr
110011110 11CC1111R
110010000 1 ICC 1RRRR
100011100 1CCC111RR
100000000 1CCC RRRRR
000011000 CCCC11RRR
000000001 CCCC1RRRR
000010000 CCCCCCCC1
therefore G(A(N£)) is an interval bigraph.
The principle behind an ATE generalizes and the generalization is what
prohibits the bipartite graphs derived from Mi and Ni from being IBGs. This
structure was identified and defined by Feder, Hell, and Huang in [12] as a
generalization, or perhaps a modification, of the principle behind the asteroids
defined by Gallai in [14] that helped him characterize transitively orientable
graphs. We will not define Gallais asteroids, but we note that we will deal with
3-asteroids, a.k.a. asteroidal triples, when we turn our attention to unit probe
interval graphs and unit interval ^-graphs. Let G = (X, Y, E) be a bipartite
graph. An edge asteroid in G is a set of 2k + 1 edges e0, ei,..., e2fc and 2k + 1
55


paths Po,i,Pi,2, Pikfi where each P^i+i joins e; to ei+i and contains and
ej+i, so that for each % = 0,1,...,2k the path Pi+k,i+k+i does not intersect
N(ei), and subscripts are modulo 2k + 1. If an edge asteroid consists of 2k + 1
edges, we call it a 2k + 1-edge asteroid. In [12] Feder, Hell, and Huang give a
new characterization of 2-clique circular arc graphs as precisely those 2-clique
graphs that have no induced cycle of length 6 or greater and no edge-asteroid
in their complements.
For examples of edge asteroids, we refer to the graphs Mi,M2 and iV2 of
Figure 3.10. Mi has a 5-edge asteroid because there is a path between e2 and
e3 that avoids the neighborhood of e0; there is a path between e3 and e4 that
avoids the neighborhood of ei; we can find a P4)0 that avoids IV(e2); there exists
P0,i avoiding lV(e3); and finally, there is a path from e4 to e2 that avoids IV(e4).
Now, in M2 and in IV2 there is a 7-edge asteroid. So, for i = 0,1,..., 6, using the
labeling in the figure and noting that, in the parlance of the definition above,
k = 3, there is a path from e* + 3 to ei+3+1 that avoids N(ei).
There is a subtle difference between an ATE and a 3-edge asteroid, and it
can be seen using a chordless 6-cycle. Take every other edge of a 6-cycle, call
them e0, ei, e2. These edges constitute an ATE. Now, for i = 0,1,2 and taking
subscripts modulo 3, the definition of a 3-edge asteroid requires that the path
Pi+i,i+2 from ei+i to ei+2 include ei+i and ei+2, and so Nfa) n Pi+i,i+2 # 0-
Hence, a 3-edge asteroid is an ATE, but not always conversely.
Every family of Figure 3.9 generates a bipartite graph with either an ATE
or an edge asteroid, see [12], The families Wi and Dj for i > 1 have ATEs
while Mi and Ni for i > 1 have edge asteroids: M4 and Ni each have a 5-edge
56


asteroid; M2 and N2 each have a 7-edge asteroid, and so on. Essentially Mj or
Ni is obtained from M*_i or iVj_i, respectively of course, by adding 4 vertices
that induce 2 edges that become members of the (2(i + 1) + l)-edge asteroid in
Mi or Ni.
Theorem 3.8 and Theorem 3,6 imply that the determination of the complete
list of forbidden subgraphs for IBGs rests on the determination of characterizing
those bipartite graphs whose complements are 2-clique circular arc graphs, but
require 2 arcs to cover the host circle in any representation. To this end the
classes of bipartite graphs called insects are defined in [37], and bugs are defined
in [24]. An insect is any graph isomorphic to one of the three in Figure 3.12.
The bugs are given in Figure 3.13. In [37], Muller conjectured that a bipartite
graph is an IBG if and only if it has no ATE, and no induced insect. The
connection between IBGs and circular arc graphs was not known at the time of
Mullers conjecture. Indeed, the graph Ni in Figure 3.10 has no chordless cycle
of length greater than 4, no ATE, and no induced insect, but is not an IBG by
Theorems 3.8 and 3.6 (b.) and so the conjecture is disproved. However, it is
easy to verify that the complement of an insect is a 2-clique circular arc graph,
but any representation requires two arcs to cover the host circle, see [24],
We present a definition, or a structure if you will, that unifies the insects
and bugs. It was defined by Hell and Huang in [24]. Let G (X,Y,E) be
a bipartite graph. For vertices xi,x2 of G, we say that N(xi) and N(x2) are
comparable if either N(xi) C N(x2) or N(x2) C N(xi). An exobiclique of G is
a biclique induced on M C X and N C.Y such that X\M and Y\N each con-
tain three vertices with incomparable neighborhoods contained in the biclique.
57


Figure 3.13: The bugs.
58


Figure 3.14: The exobiclique of a bug. Black vertices represent X.
Figure 3.14 illustrates this definition: N(x),N(y),N(z) and N(u),N(y),N(w)
are incomparable. The vertices of degree one and two in each graph of Figure
3.12 constitute exobicliques, as do the vertices of degree two and three in each
graph of Figure 3.13.
Theorem 3.9 (Hell, Huang, 2003 [24]) If a bipartite graph G contains an exo-
biclique, then in any representation ofG by circular arcs, there are two arcs that
together cover the whole circle, and hence G is not an IBG.
3.3.1 Concluding Remarks
The most prescient finding in all of the recent work on interval bigraphs
is their connection with circular arc graphs. Indeed, this mode of exploration
has not been exhausted, at least not by the author. In particular, the results
for circular arc graphs suggest the idea of characterizing IBGs via their com-
plements. Sometimes a characterization is easier to produce or discover by way
of considering the complementary situation. The result of Trotter and Moore
59


above, and the work of Gallai on transitively orientable graphs, shows that it is
sometimes advantageous to proceed in this fashion.
In regards to circular arc graphs, we will see that they are in fact intimately
tied to IBGs and all of the subclasses of IBGs we consider in this thesis. Namely,
we will see that proper circular arc graphs correspond precisely to the comple-
ments of unit interval bigraphs, and that the interval-point bigraphs are the
complements of another class of circular arc graphs.
In summary: The IBG forbidden subgraph characterization remains an open
problem.
60


4. Interval Point Bigraphs
Suppose G is an interval bigraph with the property that, in some represen-
tation, one of the families of intervals is restricted to consist of points only; call
G an interval point bigraph. We investigate interval point bigraphs, characterize
them via the structure of their reduced adjacency matrices, show that they are
precisely the complements of a certain class of 2-clique circular arc graphs in the
fashion of the result for interval bigraphs by Hell and Huang, and identify them
with a class of bipartite probe interval graphs. We also develop a conjecture for
a forbidden subgraph characterization that is supported by the result of Tucker
in [45].
4.1 Background
Let G = (X,Y,E) be an interval bigraph and (lx,Ty) its interval repre-
sentation. We investigate what happens when we restrict one of the families of
intervals Xx,Ey to be points only. Formally, we define an interval point bigraph
to be the bipartite intersection graph G = (X, Y, E) with one partite set in
one-to-one correspondence with a collection of points, V = {pu}, and the other
partite set in one-to-one correspondence with a family of intervals I = {Iv} with
uv I?-whenever pu G Iv. Clearly, any interval point bigraph is an interval bi-
graph. We note at the outset that if G = (X, Y, E) is an interval point bigraph,
then the choice of which partite set is to correspond to a collection of points de-
pends on the structure of the graph. We make some notation to emphasize this
issue. First of all, for the purposes of this chapter, we think of a bipartite graph
61


G = (X, Y, E) as an ordered triple in which X is in the first slot, Y in the second,
and E in the third. The partite set that corresponds to the collection of points
is called the point partition. So, if G = (X, Y, E) is an interval point bigraph
and X is the point partition, this is denoted G = (V,I,E), and linguistically
by G is a PIBG (PIBG is a mnemonic for point interval bigraph); similarly,
with Y the point partition, we have G = (T,V,E), and linguistically G is an
IPBG. We use interval point bigraph if we wish to make no specification as
to which of X, Y is the point partition, that is, if we wish to speak about the
class of all bipartite graphs that are either IPBGs or PIBGs. The collection of
points and intervals representing an IPBG or a PIBG is called the interval point
representation. Figure 4.1 shows three bipartite graphs each with an interval
point representation to its right.
Recall that a directed graph is an interval digraph if two intervals S (x) and
T{x) of R can be assigned to each vertex x such that u y v if and only if S(u) n
T(v) 7^ 0. An interval point digraph is an interval digraph with T(x) a singleton
point for each vertex x. In [10] interval point digraphs were characterized via
their adjacency matrices: D is an interval point digraph if and only if A(D)
has the consecutive ls property for rows, where A(D) = (a^j), with ay = 1
if Vi > Vj and 0 otherwise. Again, we will see that form the perspective of
bipartite graphs we are able to see more structure, than from the directed graph
perspective.
In this chapter we characterize interval point bigraphs in terms of the struc-
ture of their reduced adjacency matrices and show that they are equivalent to
a certain class of bipartite probe interval graphs. Also, along the lines of the
62


9o 10,
o-
9
8
o------o
1 2 3 4 5
10 o Hq
Hll
8 9o
o--o
1 2 3 4 5 6 7
2 7 4 10

8 2 4 6 9

10'
11'
9o
12 p
10
----o-
1 3 5 7 10
2 = 8 =
41- 1 12 =
oo
1 2 3 4 5 6 7 8
Figure 4.1: Three IPBGs, dark vertices belong to the point partition
result of Hell and Huang identifying interval bigraphs with a class of restricted
circular arc graphs, we determine which class of circular arc graphs correspond
to interval point bigraphs. Recall that a component problem to determining
whether a given graph is a probe interval graph is in determining how to par-
tition the vertices into probes and nonprobes. We will show that interval point
bigraphs correspond precisely to the class of bipartite probe interval graphs in
which the probe/nonprobe partition is also a bipartition. A list of forbidden
subgraphs is given and we conjecture that this list provides a forbidden induced
subgraph list for interval point bigraphs. Following the analogous results for
interval graphs, probe interval graphs, and interval ^-graphs, we give a consec-
utive subgraph characterization for interval point bigraphs. Finally, following
63


the analogous results for unit interval bigraphs and interval bigraphs, we show
that complements of interval point bigraphs correspond precisely to a restricted
class of 2-clique circular arc graphs.
4.2 A Matrix Characterization
Recall that if G = (X,Y,E), \X\ = m, \Y\ = n, then we represent G as a
(0,1)- m x n matrix A(G) = (a^) with a,ij = 1 if and only if xiyj G E, and
throughout this thesis, rows of A{G) are indexed by X (the partite set in the
first slot). A (0, l)-matrix M has the consecutive ls property for columns if the
rows can be permuted so that the ls in every column appear consecutively; the
consecutive 1 s property for rows is defined similarly.
Theorem 4.1 A bipartite graph G = (X,Y,E) is an interval point bigraph if
and only if its reduced adjacency matrix has the consecutive 1 s property for rows
or for columns.
Proof: Let M = (m^) be an m x n- (0, l)-matrix with the consecutive ls
property for rows or for columns. For the sake of argument, we will assume that
M is found to have the consecutive ls property for rows, since the other case
for columns is symmetric. With M exhibiting consecutive ls in the rows, we
will build an interval point representation for the bipartite graph G(M) that M
represents. Call the n vertices corresponding to columns {yi,... ,yn} = Y and
the m vertices corresponding to rows {aq,..., xm} = X. Create a set of points
"P = (p(j/i), >Kl/n)} corresponding to Y with p(y1) < p{y2) < < p{yn).
Create a collection of intervals I = {I{xi),..., I(xm)} corresponding to X with
I(xi) = [l(xi),r(xi)], l(xi) = min{j : ra^- = 1)}, and r{xf) max{j : m- 1}.
This yields G(M) = (X,Y,E) with Xiyj G E if and only if p{yf) G Iixf) since
64


M has the consecutive ls property for rows. G(M) is an IPBG, as desired.
Now, let G = (X,Y,E) = (V,1,E) be a PIBG. We will show that
A(G) = (aij) has the consecutive ls property for columns; that A(G) has the
consecutive ls property for rows if G is an IPBG follows from a similar argu-
ment. We may assume that the points of V are distinct in the representation,
so let V {p(xi),... ,p(xm)} where the indexing is done so that p(xi) < p{xj)
when i < j. In A(G) make row i correspond to Xi. Because each pj G Y is
represented by an interval, /(%), the neighborhood of any pj must be a con-
secutive set with respect to the order given by the point representation. Make
column j of A{G) correspond to pj and observe entry (i,j) = 1 if p(xi) G I(pj).
A(G) exhibits consecutive ls in its columns because if p(xa),p(xc) G I(pj),
then p(xt) G I{pj), for a < b < c; that is, if aaj = acj = 1, then abj = 1 for any
a In Figure 4.1 the graph called Hll is an IPBG if we think of the darkened
vertices as being the point partition, but if we switch the roles of the partite sets,
there is no way to render an interval point representation. This anomaly distin-
guishes the class of interval point bigraphs, that is, the class IPBGs U PIBGs,
from the class of graphs that Tucker studied while determining a structural char-
acterization for the consecutive ls property for columns. We discuss this in the
next section, and proceed toward a forbidden induced subgraph conjecture for
interval point bigraphs.
4.3 Forbidden Subgraph Conjecture
A bipartite graph G = (X,Y,E) is X-consecutive if the vertices of X can
65


be ordered xa < Xb for indexes a < b such that for each vertex y G Y, N(y)
is a consecutive set with respect to <; i.e., if xay G E and xcy G E, then
xi,y G E for any a < b < c. AY-consecutive bipartite graph is defined similarly.
Clearly, if G is X-consecutive, then A(G) has the consecutive ls property for
columns; similarly, if G is T-consecutive, then A{G) has the consecutive ls
property for rows. In [45] Tucker determined what is necessary and sufficient
for a (0, l)-matrix to have the consecutive ls property for columns. It could
be said that he studied this problem because of the frequency a consecutive
ordering anomaly comes to bear in many problems, especially with intersection
graphs having interval models. Indeed, we have many instances in this thesis in
which a consecutive ordering of some collection of subgraphs is either necessary
or sufficient or both. Another structural property that is apparently important
in the study of intersection graphs of interval sort is whether the graphs in
question contain asteroidal triples. Recall that an asteroidal triple (AT) of a
graph is a set of three vertices with a path between any two that avoids the
neighborhood of the third. The next theorem of Tuckers shows why we must
choose the darkened vertices of Hll in Figure 4.1 as the point partition, and
relates ATs to consecutive order properties.
Theorem 4.2 (Tucker, 1972, [45]) A bipartite graph G = (X, Y,E) is X-
consecutive if and only if it has no asteroidal triple of G wholly contained in
X.
In Hll, the sets {1,10, 5} and {3,11, 7} are ATs and both sets are contained in
one of the partite sets, and that partite set cannot correspond to points in an
interval point representation. Thus, the problem of determining whether a given
66


bipartite graph is an interval point bigraph lies, in part, in determining which set
to use as the point partition. Tuckers result, Theorem 4.2, gives the following
theorem which is prescient to our classification of interval point bigraphs.
Corollary 4.3 A bipartite graph G = (X, Y, E) is an interval point bigraph if
and only if there is no AT of G contained in X or there is no AT ofG contained
in Y.
Proof: If G = (X,Y,E) has no AT in X, then it is X-consecutive by
Theorem 4.2, and so A(G) has consecutive ls in columns and so G is a PIBG
by Theorem 4.1. Similarly, if G has no AT in Y, then G is Y-consecutive and
A(G) has consecutive ls in rows; so, G is an IPBG.
If G is a PIBG, A{G) has consecutive ls in rows by Theorem 4.1 and G is
therefore Y-consecutive. Similarly, if G is an IPBG, then A(G) has consecutive
ls in the columns; therefore, G is X-consecutive.
So, if we were to determine all the minimal bipartite graphs that have an as-
teroidal triple in each of the partite sets, then we would have a complete list of
forbidden induced subgraphs for interval point bigraphs.
The way Tucker characterized the consecutive ls property for columns in
(0, l)-matrices was by looking at the bipartite graphs that represent the matrices.
In our terminology, he characterized the X-consecutive bigraphs by forbidden
subgraphs via Theorem 4.2. Here is his result.
Theorem 4.4 (Tucker, [45]) A bipartite graph G = (X,Y,E) is X-consecutive
if and only if it has no subgraph isomorphic to any of the graphs in Figure f.2,
67


where darkened vertices represent X.
Note that there are three infinite families in Figure 4.2: C(n),IIn, and IIIn,
where C(n) is simply all cycles with even length greater than 4; and we have
given examples of Hi, //2, J/3 and III\, III2, Ilh and Ilh. Also note that II\
and IIIn for n < 3 are IPBGs, so this list does not serve as a complete forbidden
subgraph list for interval point bigraphs. To see that III, IIIi, Ilh, and Ilh
are IPBGs, refer to Figure 4.3 and see that each has a reduced adjacency matrix
with consecutive ls in the rows. Referring to Figure 4.2, Ilh is not an interval
point bigraph because the sets {x, y, z} and {c, d, e} are ATs, and observe in II2
and //3 the sets {a;, y, z} and {a, 1,2} are ATs.
Next we give our conjecture for a complete list of forbidden induced sub-
graphs for interval point bigraphs. The necessity of the conjecture is easy to
verify: simply locate an asteroidal triple in each partite set.
Conjecture 4.5 A bipartite graph G is an interval point bigraph if and only if
G has no induced subgraph isomorphic to any in Figure 4.4, or C(n),IIIn (for
n > 4), IIn (for n > 2), III, or ELIO of Figure 4-2.
4.4 A Consecutive Order Characterization
Recall that a collection Q = {G\,... ,Gm} of subgraphs of a graph G is
consecutively ordered if v G Gi n Gk => v e Gj for G{ < Gj < Gk- If for
each e e E(G), e G E(Gi), for some Gi G Q, then Q is a cover for E(G), or
Q covers E(G)\ if there is exactly one Gi containing e for each e G E(G), then
Q is a partition for E(G), or Q partitions E{G). We have characterizations for
interval /c-graphs via a cover of consecutively orderable complete multipartite
68


x= { }
Figure 4.2: Forbidden X-consecutive bigraphs.
subgraphs, and in particular, a characterization for interval bigraphs via a cover
of consecutively orderable bicliques. Also, we have a characterization for probe
interval graphs via a consecutively orderable cover of maximal quasi cliques,
69


X
d b a c
ih
a
b
z
C d
^0110
y 1 1 0 0
z 0 0 1 1
w 1 1 1 1
III 1 * a b C
T X i 0 0
a A y 0 1 0
T Z 0 0 1
I <>- i w 1 1 1
y b vv c
abed
M 1 0 0
y 1 0 0 0
z 0 0 0 1
11110
2 0 111
b c
a b d c e
x 0 0 10 0
y ioooo
Z 0 0 0 0 1
1 1110 0
2 0 1110
3 0 0 1 1 1
Figure 4.3: IPBGs from Tuckers list.
given by Theorem 2.4. Recall that a maximal quasi clique Q = (Pq,N0) in a
probe interval graph H is a subset P0 C V(H) of probes that form an induced
complete subgraph of H, together with a (possibly empty) set N0 cV(H) \ P0
of nonprobes, each adjacent to every probe in P0. Our next result gives a
characterization for interval point bigraphs in terms of a consecutive ordering of
stars that form a partition of E(G), and hence indicates the distinction between
interval point bigraphs and interval bigraphs. A star is a AT1>n, for n > 0, and
the center of a star is the partite set of size 1 (for a Ki^ either vertex may be
thought of as the center).
70


;o
2 n
1
2n
n S5 0
Figure 4.4: Forbidden interval point bigraphs; dashed edges may or may not
be present.
Theorem 4.6 A bipartite graph G = (X, Y, E) is an interval point bigraph if
and only if it has a consecutively orderable edge partition of stars with all centers
in the same partite set.
71


Proof: Let G = (X,Y,E) be an IPBG; the case for G a PIBG is similar.
By Theorem 4.1, A(G) = (a*,-) exhibits the consecutive ls property for rows.
Let ai be the first column in which a 1 appears in row i. Permute rows of A(G)
so that {oj} forms a nondecreasing sequence. Now, with yj £ Y and x^ £ X
corresponding to column j and row i, respectively, take each yj as the center of
star Sj and put Sj {yj}U{xi : Xiyj £ E}. That is, take the star given by each
column of the matrix A(G) for which {c^} forms a nondecreasing sequence; the
order of the columns gives the order of the stars. To see that this ordering is
consecutive, it suffices to check that Xi £ Sa n Sc =>- Xi £ Si for any a If Xi £ Sa n Sc, but Xi £ Sb, then aia = 1 = aac, but = 0, contradicting the
fact that A(G) has consecutive ls in the rows.
Conversely, let S = { stars with indexing corresponding to the consecutive order and so that each star
has its center in X. We will show that G is a PIBG, but note that if the centers all
belonged to Y, then we would obtain an IPBG from the analogue of the follow-
ing construction. Make a collection of points V = p(xx) < p(x2) < < p{xr),
where X{ is the center of star S,. Now, for each y £ Y, make I(y) = [l(y),r(y)\,
where l(y) = min{z: y £ Si} and r(y) = max{i : y £ Si}. We have p{xi) £ I{y)
if and only if Xi and y are both contained in some star together which happens
only if Xiy £ E. Thus, the collection of intervals and points is an interval point
representation for G.
Theorem 4.6 indicates that what distinguishes an interval point bigraph, and
hence, as we will see, certain probe interval graphs and the complements of
72


certain 2-clique circular arc graphs, from an interval bigraph is precisely the
ability to partition the edges of the graph into stars as opposed to needing to use
4-cycles and larger bicliques to cover its edges. An open and perhaps interesting
problem would be to characterize those bipartite graphs in which one is forced to
repeat edges in any consecutive cover, and hence cannot consecutively partition
the graph.
4.5 Probe Interval Graphs and Interval Point Bigraphs
The problem of choosing which partite set is to correspond to points, if in
fact the graph being considered is an interval point bigraph, is similar to the
problem of choosing how to partition vertices into probes and nonprobes when
determining whether a given graph is a probe interval graph. If the partition
into probes and nonprobes is given, then the recognition problem is solved easily,
that is, there are recognition algorithms that have polynomial running times,
see [28]. But when the partition is not given, the complexity of the recognition
problem is not known. As a simple example illustrating the issue of partition
choice, and for an example that speaks to our next results, consider H12 of Figure
4.5. In a probe interval representation, vertices c and f must be nonprobes, but
they belong to different partite sets in the bipartition of H12. Also, H12 is
not an interval point bigraph because {a, k, e} and {d, l, h} are both ATs, each
belonging to a different partite set.
The next result, especially in light of Theorem 2.6 and Theorem 4.6, gives more
precision to where bipartite probe interval graphs lie with respect to interval
point bigraphs and interval bigraphs.
73


kO 1 Q
H12
Figure 4.5: Example of a bipartite PIG with P, N partition not a bipartition.
Theorem 4.7 If G is a bipartite probe interval graph, then G has a consecu-
tively orderable partition of stars.
Proof: Let G = (P, N, E) be a bipartite probe interval graph. Theorem
2.7 shows that G is an interval bigraph. By Theorem 2.4, G has an edge cover
maximal quasi cliques that are consecutively orderable. By definition, any max-
imal quasi clique of G is either a K2 C P or a star with center in P and all other
vertices in N. Hence, the complete set of maximal quasi cliques Q consists of
stars, say Q = {Si,..., 5m}, where the indexing gives the consecutive order. By
Lemma 2.5, Q can be made so that each maximal clique of G is in exactly one
Qi G Q. Hence, Si n Si+i cannot contain an edge, since an edge is a maximal
clique in a bipartite graph. Therefore Q forms a consecutive edge partitoin of
stars.
The converse of Theorem 4.7 is not true: consider H10 in Figure 4.6. H10 is
not a PIG because, after creating an interval representation for a longest path,
we see that if either of the vertices of degree three is a probe, there is no way
to place the vertex at distance two from the path. But H10 has a consecutive
74


partition consisting of stars.
3
Figure 4.6: H10 has a consecutive partition into stars, but is not a probe
interval graph.
Corollary 4.8 An interval point bigraph is a bipartite probe interval graph, but
the converse is not necessarily true.
Proof: If G is an interval point bigraph, then by Theorem 4.6 there is
a partition of stars with all centers in one partite set. By defining P and N
to be the centers and non-centers of the stars, respectively, we obtain a com-
plete set of maximal quasi cliques of G that are consecutively ordered. H12 of
Figure 4.5 is a bipartite probe interval graph, but not an interval point bigraph.
To summarize these containment relationships just mentioned, we have
IPBGs U PIBGs C biPIGs C IBGs
with H12 and H10 as separating examples.
Next, we characterize those bipartite probe interval graphs in which the
partition of vertices into probes and nonprobes corresponds to a bipartition.
Also, given the appearance of more and more papers about probe interval graphs
75


in the literature, it gives more motivation for determining the truth of Conjecture
4.5.
Theorem 4.9 G = (X,Y,E) is a bipartite probe interval graph in which the
probe and nonprobe assignment can correspond to the bipartition if and only if
G is an interval point bigraph.
Proof: Let G = (X, Y, E) be a PIBG. If we make all vertices in X probes,
when X is the point partition and all other vertices nonprobes, we get G =
(P,N,E) is a probe interval graph. Similarly, if G is an IPBG, then we put
N = X,P = Y.
Let G = (P, N, E) be a bipartite probe interval graph with P, N each corre-
sponding to a partite set. Let {pi}^ = P, {IPi} be the family of intervals corre-
sponding to P, and {Inj} be the family of intervals corresponding to N = {nj}.
Since P is independent, for any i / j combination, IPi fl IPj = 0. Label {pi}^
so that l(pi) < l(pj) if and only if i < j. Now, for each nj, extend Inj so that
l(nj) = l(pi) for the smallest i such that pi G N(rij). Now shrink each IPi to
its left-endpoint and get G is an interval point bigraph in which P becomes the
point-partition.
4.6 Interval-Point Bigraphs and Circular Arc Graphs
Recall that a circular arc graph is the intersection graph of arcs of a circle.
The circle from which the arcs are obtained is called the host circle. A graph is a
2-clique graph if its vertices can be partitioned into two cliques and all edges not
contained in these cliques lie between them; i.e., the complement of a bipartite
76


graph is a 2-clique graph. Interval bigraphs correspond to the complements of
2-clique circular arc graphs with the property that, in some representation, no
two arcs cover the host circle. We explore which circular arc graphs correspond
to the complements of interval point bigraphs. Clearly, the circular arc graphs
we seek are 2-clique graphs.
Before we prove the main result of this section, we will use the properties of
the reduced adjacency matrices of interval point bigraphs and the vertex ordering
properties given by this characterization to characterize the complements of
interval point bigraphs. Another way to state Theorem 4.1 is as follows. A
bipartite graph G = (X, Y, E) is an interval point bigraph if X or Y can be
ordered with < such that (if X is ordered) ux,uz G E =4> uy G E for
u G Y,x,y,z G X and x < y < z (switch roles of X and Y if Y is ordered). Given
a graph G and H C.V(G), NH(v) denotes the neighborhood of v restricted to the
set H. For a graph G = (V, E), and X = {aq, £2, .., xm} C V, X is circularly
indexed if for each vinV \X, and i < j, Nx{v) is either Xi,xi+i,... ,Xj or
Xj,xj.|_i,... ,xm,xi,... ,Xi. A matrix M has the circular 1 s property for columns
if the rows can be permuted so that the ls in each column are circular, that is,
they appear in a circularly consecutive fashion; as if the matrix were wrapped
around a cylinder. The circular 1 s property for rows is defined similarly. Let us
say that the 2-clique graphs in which one of the cliques can be circularly indexed
have the circular indexing property, or the CIP. Our last and main result of this
section, we use Theorem 4.1 and the interval point represntation to show that
2-clique graphs with the CIP are circular arc graphs, and that the complements
of interval point bigraphs are 2-clique circular arc graphs with the CIP.
77


Theorem 4.10 A bipartite graph G = (X,Y,E) is an interval point bigraph if
and only if G is a 2-clique circular arc graph with the CIP.
Proof: Suppose G = (X, Y, E) is an IPBG with \Y\ n; we will prove that
G is a 2-clique circular arc graph such that Y can be circularly indexed in G. If G
were a PIBG, then G would turn out to be a 2-clique circular arc graph in which
X can be circularly indexed. Since G is an IPBG, it is F-consecutive and so we
may assume that an arbitrarily chosen iGlis adjacent to Pi,... yk G Y in G,
for 1 < i < k < n, and hence, x is adjacent to Pk+i, yn, Vu Ui-i in G. We
construct a circular arc representation for G. Let C be a circle with two specified,
diametrically opposed points p and q, with A (respectively B) the segment of
C extending clockwise from p to q (respectively q to p). Let X be the interval
point represetnation for G. Gs structure dictates that pyi < py2 < < pyn, we
may use I{x) = \pi,pk\, and we assume that the points are spaced equidistantly
by some constant, say e, and that the total width of X is pyn pyi; that is, the
leftmost interval has left endpoint equal to pyi and the rightmost interval has
right endpoint equal to pVn. Place a copy of X in A with pyi = p and place a
copy of X in £ with pyi = q. We assume A and B are large enough so that
Pyn < q in A, and pVn < p in B. Let R(v) denote the arc corresponding to
vertex v and let cc(v) denote the counter-clockwise endpoint of R(v) and cl(v)
the clockwise endpoint of R(v). Construct open R(v) = (cc(v),cl(v)) for each
v G V(G) as follows. Put (cc(yi)) = pyi G A and cl{yi) = pyi G B. Put
cc(a;) = pVk G B and cl(x) = pyi G A. In this representation R(x) fl R(pj) 0
whenever j G {k + 1,..., n, 1,..., i 1}, so Y is circularly indexed. Since
x was arbitrary, this construction applied to each x G X gives a circular arc
78


representation of G, a 2-clique graph with the CIP.
Let G be a 2-clique graph with X and Y the sets inducing the cliques, and
M(G) = (rriij) the adjacency matrix for G\ that is rriij = 1 if ViVj £ E(G) and 0
otherwise. If X can be circularly indexed, then we can permute corresponding
rows and columns so that
I\X\x\X\ A
[ AT !\y\x\y\.
M(G) =
where I is the square matrix with 0s on the diagonal and ls everywhere else,
A = M(G)[X]Y], that is, A is the matrix induced on the rows corresponding to
vertices in X versus the columns corresponding to the vertices in Y. Since X is
circularly indexed, A has the circular ls property for columns, and AT has the
circular ls property for rows. Thus,
M(G)
I\X\x\X\ A
AT I
where A is the reduced adjacency matrix for the bipartite graph G and will
clearly have the consecutive ls property for columns. Hence, G is a PIBG, by
Theorem 4.1. Taking complements, and disregarding which clique can be circu-
larly indexed, we see that if G is a 2-clique graph with the CIP, then G is an
interval point bigraph.
In Figure 4.7 we have illustrated the idea behind the proof of Theorem 4.10
by constructing a circular arc representation for G using the construction in
the proof. The figure drawing environment at the disposal of the author is not
conducive to drawing arcs and circles, so we have used a square representation
- nothing is lost of course, since topologically circles and squares are the same.
79


y i yi y 3
*3 f
>2?
-o-
-o-
*i y 1 *2 ^3 *4
*1
*2
o
*3
O
X4
o
interval point
representation
for q
circular arc representation for G
Figure 4.7: A circular arc representation of the complement of an interval
point bigraph using the method in the proof of Theorem 4.10. The dashed lines
are meant to indicate the arcs.
80


We conlude with the following list of equivalences for bipartite graphs.
Theorem 4.11 Let G = (X, Y, E) be a bipartite graph. The following are equiv-
alent:
(1.) G is a bipartite probe interval graph in which the probe/nonprobe partition
can correspond to the bipartition;
(2.) G is a 2-clique circular arc graph in which one of the cliques can be circularly
indexed;
(3.) G is an interval point bigraph;
(4-) G has a consecutively orderable edge partition of stars in which all centers
belong to the same partite set.
81


5. Unit Interval ^-graphs
Results in this chapter have arisen from restricting attention to the case
where all intervals have the same length. We have found this restriction, as it
forces quite a bit of structure, to be fruitful in terms of discovering structural
properties. Unit interval bigraphs for example are shown to be equivalent to over
a dozen other classes of structured graphs and structural properties. Also, like
the unit interval graphs, see [39], and unlike tolerance graphs, see [2], unit inter-
val bigraphs are equivalent to proper interval bigraphs. The characterizations
include one of forbidden substructures, and this characterization gives a struc-
ture theorem for a class of structured (0,l)-matrices similar to the way Theorem
4.4 gives a structure theorem for the consecutive ls property for columns.
5.1 Background
We report on findings obtained while determining a forbidden subgraph
characterization for unit interval bigraphs, and unit interval /c-graphs. The
results for unit interval bigraphs involve several other classes of graphs some of
which we now define. Some definitions however are not given in this background
section, but are defined closer to where they are needed.
Recall an interval k-graph (IkG) is the ^-partite intersection graph of k dis-
tinct families of intervals of R, T = {Zi,Z2,..., Tk}, with vertices adjacent if
and only if their corresponding intervals intersect, and each interval belongs to
a distinct family. An IkG is a unit interval k-graph (UlkG) if in some repre-
sentation all intervals can be made to have identical length. For G an IkG, or
82


UlkG, recall from Chapter 2 that the interval classes correspond to independent
sets of G and are essentially obtained from a proper coloring of G. We also re-
marked in Chapter 2 that not every proper coloring is conducive to an interval
^-representation, for example giving every vertex a distinct color works only if
G is an interval graph. But we also showed that IkGs are perfect and in do-
ing this showed that using u>(G) interval classes will always suffice. But within
the class of UlkGs, we have found anomalous occurrences in which only a very
specific w(G)-coloring corresponds to an interval class assignment conducive to
a unit interval ^-representation. Figure 5.1 serves as an example of this. The
graph of Figure 5.1 is a UlkG, as evinced by the top representation, but with
the bottom interval class assignment, there is no way to place 1(6) so that it
does not intersect 1(2) or /(4), which violates the fact that vertices 2 and 6,
and 2 and 4 are not adjacent. Thus, a characterization of UlkGs will require
the understanding of this and perhaps the invention of a very specific coloring
scheme. The example suggests that the scheme must take into account not only
the neighbors of a vertex, but also the neighbors at distance 2.
If no interval properly contains another, in some interval /c-representation,
then the graph corresponding to the representation is a proper interval k-graph.
The vast majority of this chapter is devoted to the bipartite unit interval k-
graphs, that is, the unit interval bigraphs (henceforth UIBGs), and the proper
interval bigraphs (henceforth PIBGs). We will now define a few of the notions
and classes of graphs we will explore.
The following class of bipartite graphs turns out to be equivalent to the
class of unit interval bigraphs, and has been useful in giving the authors helpful
83


1
1
2
o
3
4
-
6 7
5
2
16
3
5
1 2 3 4 5
--------------
6 7
1 ; 3
2'
7
4;
5
Figure 5.1: A UlkG in which the ability to represent it rests on using a
particular interval class assignment. The horizontal dashed lines distinguish the
interval classes (as do their shadings), the vertical dotted lines serve to help
determine intersection.
perspectives. A valuation bigraph, G = (X,Y,E), is a bipartite graph, with
functions f : X > R, g :Y > R such that if x G X, y 6 Y, then xy E E if and
only if |f{x) g{y)\ < 1. The functions f,g are called the valuation functions.
In other words, a valuation bigraph is the bipartite analogue of a valuation k-
graph discussed in Chater 1. We discuss valuation ^-graphs again briefly as
a prelude to Theorem 5.1, and reiterate that the bipartite graph perspective,
and in particular the perspective given by valuation bigraphs, has proved to be
fruitful for us.
An asteroidal triple (AT) is a set of three vertices with a path between any
two that does not intersect the neighborhood of the third. A graph that has
no asteroidal triple in any induced subgraph could be thought of as a graph
that grows in only two directions. The interval graphs have been shown to be
equivalent to the class of graphs that do not contain an asteroidal triple on
84


any induced subgraph, or an induced cycle of length at least four, see [4]. AT-
free graphs have been studied extensively perhaps because of the appearance
of ATs as forbidden substructures in many classes of graphs; they have been
characterized in [9]. It will turn out that all of the bipartite graph classes we
investigate in this chapter are precisely those bipartite graphs that are AT-free.
The class of graphs we define next is not always thought of as a class of
intersection graphs, but we present the definition which renders them as such.
A permutation graph is the intersection graph of a family of line segments whose
endpoints lie on two distinct parallel lines; vertices are adjacent if and only if
their corresponding line segments cross. For the alternative non-intersection
model, let tt be a permutation of {1,2, ...,n}, and consider the list [7r (1),
7t(2), ..., 7r(n)]. Let G = (V, E) have vertices V = {ui, v2,..., vn} with ViVj E E
whenever i < j and % is to the right of j in the list; then G is a permutation
graph.
A comparability graph is a graph G = (V., E) in which E may be oriented so
that for x, y, z E V if x > y and y z, then x > z; that is E has a transitive
orientation. A cocomparability graph is a graph whose complements edges have
a transitive orientation.
5.2 Characterizations
Originally, valuation bigraphs were conceived in a more general sense in
order to model simultaneous rankings, or quantifiable judgments, placed on a
collection T of (not necessarily distinct) sets. If the functions fi, ,fk represent
the quantifiable criteria placed on T = {Xl, ... ,Xsuch that fi : Xi > M,
and we wish to examine the rankings under the given criteria with respect to
85


some tolerance e > 0, it may make sense to examine the A;-partite graph G =
{E,E) with uv G E \fi{u) fj(v)\ < e, where u G Xi,v G Xj. We call a
graph having such a representation a valuation k-graph, and the specialization
to E = {X,Y} are the valuation bigraphs. The next result clearly specializes
to valuation bigraphs.
Theorem 5.1 A graph G is a valuation k-graph if and only if it is a unit interval
k-graph.
Proof: Let G = (X1;... ,Xk,E) be a valuation ft-graph with valuation
functions fi : X{ > K, for i = 1,..., k. Create an interval I(v) = [l(v),r(v)]
for each vertex v of G as follows: put I(v) = [fiiy) |, fi(v) + |] for v G Xi,
and say I(v) belongs to interval class 2j. Note that if two vertices are in the
same partite set, then they are not adjacent by definition. So, with x G Xi and
y G Xj we show that xy G E if and only if I(x) n I(y) ^0. We have
I Mx) fAy) | < i ^
-1 < fi(x) fj(y) < 1 &
-1 < fi{x) + ^ fj(y) ^ < 1 &
1 < r(x) r(y) < 1
Ir(x) r(y) \ < 1 <£>
I(x) n/(j/) / 0,
where the last equivalence follows since all intervals have length 1. Thus, if G
is a valuation /c-graph, then G is a unit interval /c-graph.
Conversely, suppose that G = {Xi,..., Xk} E) is a unit interval &-graph with
interval representation T = {X;,... ,Zfc}, where \I(v)\ = 1 for each I(v) G E, and
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I I VARIATIONS ON INTERVAL GRAPHS by David E Brown B.S., Metropolitan State College of Denver, 1999 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics 2004

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This thesis for the Doctor of Philosophy degree by David E Brown has been approved by Ellen Gethner

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Brown, David E (Ph.D., Applied Mathematics) Variations On Interval Graphs Thesis directed by Professor J. Richard Lundgren ABSTRACT Interval graphs are a well-known type of intersection graph whose invention or discovery is credited to Benzer in 1959, in the course of his studies of the topology of the gene, and sometimes to Hajos in 1957, with respect to his purely combinatorial question that asks basically which graphs have a representation by a set of intervals so that vertices are adjacent if and only if their corresponding intervals intersect. Interval graphs have been studied extensively since then and many nice properties have been found to be associated with them; from properties relating to structure to properties that admit much applicability. Motivated by a problem loosely related to Benzer's problem, Zhang created probe interval graphs to model a specific problem in the physical mapping of DNA. A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes and intervals can be assigned to vertices so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of the vertices is a probe. Whether or not the vertex partition is given is a factor when determining if a given graph is a probe interval graph. In 1984 Harary, Kabell, and McMorris introduced interval bigraphs as a natural iii

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extension of the interval graph idea. An interval bigraph is a bipartite graph with an interval assigned to each vertex, and vertices are adjacent if and only if their corresponding intervals intersect and belong to a distinct partite set. A talk given by the author sparked a question that lead to the invention of interval k graphs. An interval k-graph is a k-partite graph with an interval assigned to each vertex with vertices adjacent if and only if their corresponding intervals intersect and each vertex belongs to a distinct partite set; i.e., an interval bigraph is an interval k-graph with k = 2. In this thesis it is shown that any probe interval graph is an interval k-graph and this result is capitalized upon especially in the bipartite case. Specifically, a necessary condition on the existence of a consecutively orderable biclique partition is shown to hold for bipartite probe interval graphs. Also, the bipartite probe interval graphs in which the probe/nonprobe partition can be made to correspond to a bipartition are precisely the interval-point bigraphs and a list of forbidden subgraphs is conjectured. An interval-point bigraph is an interval bigraph in which the intervals corresponding to vertices of one of the partite sets can be reduced to points. Interval point bigraphs (and the probe interval graphs to which they correspond) are characterized by properties of their re duced adjacency matrices, as are unit interval bigraphs and interval bigraphs A unit interval bigraph is an interval bigraph in which all intervals have the same length, a proper interval bigraph is an interval bigraph in which no inter val contains another properly. Unit interval bigraphs are characterized in several ways. They turn out to be precisely the asteroidal triple-free bipartite graphs, the bipartite cocomparability graphs (and hence the incomparability graphs of iv

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width two partial orders), bipartite permutation graphs, valuation bigraphs, and are characterized further by an ordering of their vertices and by forbidden subgraphs. Some of these characterizations were found independently by Hell and Huang, and some complement existing and newly found characterizations for interval bigraphs. Characterizations via consecutive orderings of certain subgraphs have been given for interval graphs by Gilmore and Hoffman in 1965 and for probe interval graphs by Zhang in 1994. In this thesis interval k-graphs are characterized by a consecutively orderable edge cover of complete multipartite subgraphs. In particular a bipartite graph is an interval bigraph if and only if there is an edge cover of bicliques that can be consecutively ordered. Whence the necessary condition mentioned above for bipartite probe interval graphs. Valuation k-graphs are to unit interval k-graphs as indifference graphs are to unit interval graphs. As a unit interval graph is the incomparability graph of a semi-order, a unit interval k-graph is the incomparability graph of a par tial order with width at most k. Unit probe interval graphs are shown to be cocomparability graphs in several ways and hence they are the incomparability graph of a partial order as well. Unit probe interval graphs are characterized via forbidden subgraphs in the cycle-free case, and particular their relationship with bipartite unit tolerance graphs is made precise. A circular arc graph is the intersection graph of arcs of a circle. Closer in age to interval graphs than any of the aforementioned classes of graphs, they have also been studied extensively and characterized in many ways by A Tucker, in particular. Recently Hell and Huang have discovered that interval bigraphs v

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and proper interval bigraphs each correspond precisely to the complements of a restricted class of circular arc graphs: e.g., proper interval bigraphs are precisely the complements of proper circular arc graphs. We show a similar result for interval point bigraphs, filling in the hitherto missing link in the circular arc graph/ complement of interval bigraph hierarchy. This abstract accurately represents the content of the candidate's thesis. I recommend its publication. Signed Vl

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DEDICATION I dedicate this thesis to my wife Tessa, and to my grandparents Addamae and Dave Brown They have put much more into me than I have into this thesis.

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ACKNOWLEDGMENT The time I have spent at CU-Denver has been among the best of my life. For this I have the following people to thank. I am grateful to the Graduate Committee for providing me with teaching assistantships, and to the faculty for the other honors that have been bestowed upon me, including the Lynn Batemari Fellowship and the Ph.D. fellowship. Liz and Marcia have all too often shown me what to do and when to do it, I thank them for their patience and willingness to help. I am grateful to Steve Flink for hfs company during some of the work that created this thesis, and for his tutelage in many things non-mathematical. To Rob Rostermundt and Steve Flink, thank you for the company and motivation during the Christmas "break" that we spent preparing for the analysis prelim. I am grateful to Dustin Stewart for the advice, mathematical and otherwise, he has given me during the time we have shared an office together. Thank you, Kathy, Bill, Stan, and Ellen for serving on my Ph.D. committee and for the comments that have enhanced the quality of this thesis. And with all my heart, I thank my advisor, Rich Lundgren. I know that to a great extent, my experience at UCD has been so fantastic because of his advice, guidance, and presence in general. I hope that in my future endeavors I can give to others at least some of what he has given to me.

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CONTENTS Figures . xii 1. Introduction 1 1.1 Prologue 1 1.2 Notation 2 1.3 Background 3 1.4 Evolution of my Research 5 1.4.1 Ordered Sets and Cocomparability Graphs 8 1.4.2 (0, 1)-matrices. 13 1.4.3 Circular Arc Graphs 14 1.5 Progress on IBGs 15 1.5.1 :F Restricted to Unit Intervals 16 1.6 Summary of Chapter Contents 17 2. Interval k-graphs and Probe Interval Graphs 19 2.1 Definitions and a few details . . 19 2.2 A characterization for IkGs and a containment relationship 22 2.3 Some properties of IkGs 28 2.4 Forbidden substructures 30 2.5 Applications of IkGs 33 3. Interval Bigraphs 35 3.1 Background and definitions 35 IX

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3.2 Characterizations for IBGs . . 3.3 Forbidden Substructures and Subgraphs 3.3.1 Concluding Remarks 4. Interval Point Bigraphs 4.1 Background . 4.2 A Matrix Characterization 4.3 Forbidden Subgraph Conjecture 4.4 A Consecutive Order Characterization 4.5 Probe Interval Graphs and Interval Point Bigraphs 4.6 Interval-Point Bigraphs and Circular Arc Graphs 5. Unit Interval k-graphs 5.1 Background 5.2 Characterizations 5.3 Proper IkGs 6. Unit Probe Interval Graphs 6.1 Background ....... 6.2 UPIGs are Cocomparability Graphs 6.3 Cycle-free Unit PIGs 6.4 Bipartite UPIGs 6.5 Concluding Remarks 7. Concluding Remarks and Future Directions 7.1 Interval k-graphs and Probe Interval Graphs 7.2 Interval Bigraphs .... 7.3 Interval Point Bigraphs X 39 50 59 61 61 64 65 68 73 76 82 82 85 106 113 113 115 121 131 133 135 135 138 140

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7.4 Unit Interval k-graphs ... 7.5 Unit Probe Interval Graphs 7.6 A Final Remark On Probe Interval Graphs References . . . . . Xl 140 141 142 144

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FIGURES Figure 2.1 An IkG that is not a PIG 22 2.2 An IkG and its consecutive multipartite subgraphs 27 2.3 The complement of a 6-cycle, which is not an IkG 30 2.4 Examples of ATEs . . . 31 2.5 Two minimal forbidden graphs for IkGs 32 3.1 H10 and an interval bigraph representation. 35 3.2 A zero partitionable matrix . . 37 3.3 NL10: The smallest tree that is not an IBG and the only forbidden subgraph for cycle-free IBGs . . . 40 3.4 Interval bigraph representation for the matrix A 47 3.5 An almost-insect s interval bigraph representation 48 3.6 A circular arc graph. . . . 49 3.7 Forbidden configuration for Theorem 3.6 (e). 49 3.8 Forbidden configurations for Theorem 3.6 (f). 50 3.9 The finite families of finite sets that represent the forbidden complements of 2-clique circular arc graphs. . . . . 53 3.10 Some of the corresponding graphs of the families in Figure 3.9. The thicker edges are ATEs or edge asteroids. The black vertices represent elements and the white vertices represent subsets. 3.11 The graphs corresponding to r1 r2 r3 ........ xii 54 55

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I 3.12 The three insects. : 3.13 The bugs ..... 3.14 The exobiclique of a bug. Black vertices represent X. 4.1 Three IPBGs, dark vertices belong to the point partition 4.2 Forbidden X -consecutive bigraphs. 4.3 IPBGs from Tucker's list. . 4.4 Forbidden interval point bigraphs; dashed edges may or may not be 58 58 59 63 69 70 present. . . . . . . . . 71 4.5 Example of a bipartite PIG with P, N partition not a bipartition 74 4.6 H10 has a consecutive partition into stars, but is not a probe interval graph. . . . . . . . . 75 4. 7 A circular arc representation of the complement of an interval point bigraph using the method in the proof of Theorem 4.10. The dashed lines are meant to indicate the arcs. . . . . 80 5.1 A UikG in which the ability to represent it rests on using a particular interval class assignment. The horizontal dashed lines distinguish the interval classes (as do their shadings), the vertical dotted lines serve 5.2 5.3 5.4 5.5 to help determine intersection. . . . 84 Forbidden unit interval bigraphs, include Ck, k 6 91 Forced adjacency diagram for Theorem 5.16: dashed edges are forced 100 The labeling satisfies (i), but not (i) and (ii) of Theorem 5.16 100 Forbidden UikGs that are asteroidal triple-free. . . 107 xiii

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5.6 A function representation for a proper interval 3-graph ala Theorem 5.23. Directly below H is its proper interval k-representation, and directly below that is afunction representation for H. 111 5. 7 A function representation for F: an interval k-graph that is not a proper IkG. . . . . . . . 112 5.8 A function representation, and a permutation representation for a proper interval bigraph. . . . . 112 6.1 NC7 is a tolerance graph, not a bounded tolerance graph. 118 6.2 A unit tolerance graph that is not a PIG. 119 6.3 A parallelogram graph. . . 121 6.4 A parallelogram representation for a bounded tolerance graph. 121 6.5 Forbidden partitioned cycle-free UPIGs -darkened vertices are probes.122 6.6 Labeling used in lemmas 3.1, and 3.2. 122 6.7 Forbidden PPIGs and UPIGs. . 123 6.8 Cases for an unsignedvertex v -reducible vertices are labeled r, darkened vertices have been assigned to P. 130 6.9 Ti and 1ri . . . . 6.10 Forbidden bipartite UPIGs (include F1 F2 and Hi, i 2: 0). 7.1 A parallelogram representation for a weakly proper PIG. 7.2 2-trees that are not UPIGs. .............. 7.3 Possible subgraphs induced on probes of G of Figure 7.2 7.4 Asteroidal triple-free, 3-chromatic, minimal, forbidden PIGs. The 130 131 138 142 142 principle for constructing these can be used to create infinitely many minimal forbidden PIGs. . . . . . . 143 xiv

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I. Introduction 1.1 Prologue A common problem in combinatorics is the following. Given a structural property X, determine a collection or list of objects such that, for each object of, the presence of the object precludes X, and if every object of is absent, then X is present. A list is complete relative to X, if is minimal as a list and the absence of every item on the list guarantees that X exists. The work in this thesis stems from the attempt to determine an for varieties of intersection graphs (serving as X) that have a family of intervals of the real line as their representation. Furthermore, whenever possible, we attempt to construct so that it consists of (sub )graphs. If such an is determined, we have a forbidden induced subgraph characterization Such a characterization is difficult when the list is extensive, and typically the determination of the list may only arise from ad hoc arguments. But some of the graph classes we consider turn out to have "nice" characterizations. By 'nice' we mean that their corresponding is short or easy to describe. Given a particular X, and a possibly complete list of forbidden substructures, it is relatively easy to see that the prohibition of every thing in is necessary, but proving that is complete is typically difficult, and one is usually left with proof techniques that fall short of elegance. Given some combinatorial object 0 and structural property X, the recognition problem is that of determining whether 0 has X. The certification problem ; relative to 0 and X, is that of proving that 0 does not have X. Assuming that 1

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X is important, the motivation for finding relative to X, lies partly in the fact that makes the recognition problem easy. Because, even if is large as a list and its items are complicated, the complexity of its items is relative to the size of 0 for which X is being searched. Therefore, almost always, having for X leads to fast recognition algorithms. If is not conducive to fast recognition algorithms, it may be conducive to fast certification algorithms; this is because if is known for a particular X, one need only find an object from in 0 to certify that 0 does not have X. 1.2 Notation Notation will be developed as needed, but at the outset we can say the following (now typical) notational conventions will be used. A graph G consists of a set of vertices V (or V(G)) and a collection of edges E (or E(G)) which in turn consists of distinct unordered pairs of distinct elements of V. Note: most authors call this a simple graph, we call it simply a graph. Some authors would say that a graph G is an irrefiexive, symmetric relation E on V. To denote an edge we juxtapose two vertices and say the vertices are adjacent; that is, for u, v E V, uv E E denotes u and v are adjacent or uv is an edge. Also, we may use the symbol +-+ between vertices to indicate that they are adjacent and t+t to indicate that they are not. Given a subset V' V(G), the graph induced on V' is denoted G(V') and is the subgraph in which vertices of V' are adjacent in G(V') whenever they are adjacent in G. A graph will often be denoted by an ordered pair that indicates both the vertex and edge set: G = (V, E). A graph G = (V, E) is k-partite if V can be partitioned into k subsets V = V1 U U Vk such that each edge consists of vertices from distinct 2

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partite sets of V, alternatively, G(Vi) has no edges for each i. A subset of vertices V' V(G) with G(V') having no edges is called an independent set. Much of this thesis is devoted to bipartite graphs. So, in case G is bipartite and V partitions into V = XU Y, such a graph is denoted G = (X, Y, E). A complete graph is one with every pair of vertices adjacent, while a complete multipartite graph is a k-partite graph with every pair of vertices that belong to different partite sets are adjacent. A directed graph or digraph D = (V, R) is a generalization of a graph in which R, the set of arcs consists of ordered pairs of V, and for vertices u and v if the ordered pair ( u, v) E R, then we may denote this u ---+ v. So, a digraph D = (V, R) is simply a relation R on V with no restrictions; that is, the relation could be reflexive for some elements of V and not others, and (u, v) E R does not imply (v, u) E R. It is typical to represent a graph with a drawing in which vertices are de picted by dots and if two vertices are adjacent, then a line connecting them is drawn. A digraph is typically represented in the same way, but with arrows indicating the order; in a drawing for a digraph D = (V, R) with ( u, v) E R, an arrow is drawn from u to v. In this thesis, vertices often correspond to objects, like intervals, and the correspondence will be indicated by subscripts or by functional notation. For example, I(v) may denote the interval that corresponds to vertex v, so may Iv, and if needed, l(v), r(v) denote the left and right endpoint of I(v). 1.3 Background An intersection graph is a graph G whose vertices correspond to a family of sets F and vertices are adjacent if and only if their corresponding sets in3

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tersect; :F is called a representation of G when G is the intersection graph of :F. Any graph is the intersection graph of some family of sets, see [33). But when :F is restricted or the rules that determine adjacency are modified, in teresting classes of graphs may be created The recent text [34) evinces the diversity and importance of the intersection graph perspective. Probably the most well-known of the intersection graphs is obtained when :F is restricted to be intervals of JR. In this case, the intersection graph of :F is an interval graph. Interval graphs have been extensively studied and characterized in many ways, including a characterization by forbidden substructures. The papers [4), [13), and [15) are the seminal papers on interval graphs and the techniques and char acterizations in them are easily seen to be the prescient ones for many of the analogous intersection graph classes that are chic now. The texts [34), [20), and [40) contain excellent introductions and exhibit refinements of the proofs in the aforementioned papers The interest in many classes of intersection graphs are application driven. For example, interval graphs are often attributed to an attempt to model ideas about the fine structure of the gene, see [1 J. Advances in the field of molecular biology, and genetics in particular, solicited the need for a new model. In 1994 P. Zhang introduced probe interval graphs in an attempt to aid a problem called cosmid contig mapping, a particular component of the physical mapping of DNA, see [48, 49]. A probe interval graph (PIG) is a graph G = (V, E), where V can be partitioned into (P, N), such that there is an interval corresponding to each vertex and vertices are adjacent if and only if their corresponding intervals intersect and at least one of the vertices belongs to P. PIGs generalize interval 4

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graphs (put N = 0) and are an instance of an intersection graph with a modified adjacency rule. We note that although applications often accompany mathematics as mo tivation for study, sometimes it is clear that the mathematics has strayed far from its conception as a consequence of application and is now studied for its own sake. This is what happens in this thesis. For example, interval graphs can also be attributed, and perhaps more appropriately so, to a question posed by Hajos in 1957 see [21 ). He asked, basically what graphs have a representation by collection of intervals of JR. Benzer never mentioned interval graphs as such, but the mathematical community picked up and ran with the idea. Most of the research in this thesis is driven by combinatorial questions fueled by the modi fication of a definition or the attempt to find a result analogous to a canonical one. The next section tells the story of how some of the results came to be, approximately 1.4 Evolution of my Research Most of the work in this thesis stems from an attempt to characterize PIGs. Because of their purported application to physical mapping of DNA, combined with the fact that they generalize interval graphs, the research community has shown an interest in PIGs. While investigating PIGs, Cary Miller, for reasons having to do with the biology of the application PIGs model, asked: "what if 'at least' is changed to 'exactly' in the definition of a PIG?" Our preliminary investigation showed that a distinct and interesting class of graphs is obtained from this definition. We eventually came to call this class of bipartite graphs interval bigraphs (IBGs). 5

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Due mainly to nomenclature, for quite some time we did not realize that IBGs had already been introduced and studied. While we saw IBGs as a natural extension of PIGs, IBGs were introduced in 1984 as a natural extension of interval graphs and called hi-interval graphs, see [22]. There was also an analogue to interval bigraphs in the context of digraphs introduced in 1989, see [10]. An interval digraph (ID) is a directed graph with an ordered pair of intervals (Su, Tu) corresponding to each vertex u such that u -+ v if and only if Su n Tv #-0. In [10] IDs are introduced as an analogue to interval graphs. Das et al. [10] also seemed to be unaware of the paper [22] that introduced and (erroneously) characterized hi-interval graphs. We note that the models for both IDs and IBGs are essentially the same: let each Sx correspond to a vertex x E X and each Ty correspond to a vertex y E Y and you will get an IBG G = (X, Y, E) with lXI = IYI. Therefore, results obtained from the perspective of interval bigraphs complement those for interval digraphs We note, with an admitted bias, that we believe the bipartite graph perspective to be more fruitful because of the structure present in bipartite graphs that is not present in directed graphs. An alternate definition of IBGs is the following: An IBG is the intersection graph of two different families of intervals with vertices adjacent if and only if their corresponding intervals intersect and belong to different families Note that an IBG is bipartite with each partite set corresponding to one of the two interval families. At a talk given at DIMACS by the author in the summer of 2000 on early results on IBGs, T.S. Michael asked the following question: "Given a graph, how many different interval classes are required to represent it so that vertices are adjacent if and only if their corresponding intervals intersect 6

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and belong to distinct classes?" Whence interval k-graphs (IkGs); i.e., IBGs with "two" replaced by "k". The answer to the question for some graph G, by the way, is x( G), but not all graphs can be represented in such a way. For example only a very restricted class of trees are IkGs and ann-cycle, for n 5 is not an IkG for any k. In Chapter 2 we prove that IkGs generalize PIGs, and are perfect, and we give a characterization for IkGs in terms of a consecutively orderable collection of subgraphs. However most of this thesis is focused on IBGs and subclasses of IBGs obtained by restrictions placed on one or both of the classes of intervals in the representation. In particular, at various times and for various reasons, we decided to investigate the consequences of restricting all intervals to unit length, giving the unit lEGs. At another time, we investigated what happens when one interval class corresponds to points only, giving the interval point bigraphs. We obtain very conclusive results for unit IBGs, see Chapter 5, and for interval point bigraphs, see Chapter 4. But for IkGs (k 3), even the case in which all intervals are of unit length remains open. We remark that, because of the structure imposed upon the representation, the unit case is often the most fruitful in terms of discovering structural properties. Indeed, unit IBGs turn out to be identical to many other well-known classes of bipartite graphs and the complements of others, see Chapter 5 Remark: The author contends that IkGs have more structure than PIGs and so it makes sense to investigate IkGs primarily, with an eye open for ancillary results regarding PIGs. This is the approach taken here and it has turned out to be fruitful in some cases. Many of the directions for investigations in this thesis are due to the work by 7

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earlier mathematicians who studied interval graphs and probe interval graphs and from contemporary ones studying interval bigraphs and interval digraphs. The results of Roberts, Gilmore and Hoffman, Hell and Huang, and Das, San, Sen, Roy, and West on unit interval graphs, interval graphs, interval bigraphs and circular arc graphs, and interval digraphs, respectively, have sparked many of the ideas that led to results. We thank them for their passive, but powerful contribution. The next paragraph gives some specifics to how the aforemen tioned mathematicians have motivated this research. The indifference function Roberts associated with unit interval graphs, see [40, 38], led to the idea of valuation functions in relation to interval k-graphs. The consecutive order characterizations of Gilmore and Hoffman in [15] led us to ask, whenever plausible, whether such a characterization exists for the class under consideration (this will become obvious). The vertex ordering character izations by Hell and Huang motivated a result in Chapter 5. Hell and Huang's results relating interval bigraphs to circular arc graphs in [24) motivated the characterization of interval point bigraphs in Chapter 4, identifying them with a restricted class of circular arc graphs. Of course, we have used results of many others directly and they are cited. In particular, A. Tucker's work on (0,1) matrices is cited more than once in Chapter 4, and Li Sheng's work on PIGs has been useful; in particular, her characterization by forbidden subgraphs for PIGs that are cycle-free (see [43]) suggested the work in Chapter 6. In the subsections that follow, we discuss some of the other mathematical things which have come to bear, motivate, or relate to results in this thesis. 1.4.1 Ordered Sets and Cocomparability Graphs 8

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An ordered set P = (X,<) consists of a set X and a binary relation< on X that is irreflexive (x I. x for all x E X), transitive (x < y and y < z =::::::?x < z for all x, y, z E X), and therefore asymmetric (x < y =::::::?y I. x for all x, y E X). We say that elements x, y of X are comparable if either x < y or y < x, and otherwise they are incomparable. A partially ordered set (poset) P = (X,::;) has relation::; that is reflexive (x ::; x for all x EX), antisymmetric (x ::; y andy::; x =::::::?x = y), and transitive. There are two common or natural ways to associate a graph G = (X, E) with an ordered set P = (X,<). One way is to define the edges thus: E = { xy : x < y or y < x}; this renders G the comparability graph of P. The incomparability graph of P has edges defined by E = { xy : x I. y and y I. x}. A cocomparability graph G is one in which the edges of G can be directed so that the resulting digraph is transitive. So, a cocomparability graph G = (V, E) has an ordered set on V in its complement G defined by any transitive orientation: make x < y whenever x --+ y in G. In this thesis, we have asked ourselves for every class of graphs we investigate "Does this class belong to the class of cocomparability graphs?" When the answer is affirmative, there is an order defined by the complement of any graph in that class. This order may be interesting fodder for future study. But since we care only about the orders that are defined by orienting the complements of graphs, that is, by giving each edge a unique direction, the relation created can be thought of as antisymmetric. Also, we concern ourselves only with loopless graphs, so we may either assume that in the complement there is understood to be a loop at every vertex; or we may assume the opposite and find that, for our purposes, either assumption is 9

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of no consequence. Thus, we could regard the order on the complement of a cocomparability graph as a poset. And since this, for the most part, is what the prominent authors in the literature do, we will do the same, see [18, 19, 3]. Remark: There is actually discrepancy among some authors and even among the same authors from paper to paper regarding their definition of a partially ordered set. But the discrepancies stem from pragmatic reasons, and are irrel evant in terms of the theory of partially ordered sets; that is, the well-known results from poset theory still apply given any ofthe different definitions that the author has seen. Here are some other terms we will use. A poset of width k decomposes into k disjoint linear orders; a linear order in a poset (with the conventions discussed above) is one with no incomparabilities among different elements. For example we can define a width k poset on the complement of a k-partite cocomparability graph G, since in G there will be k complete subgraphs, and a complete subgraph with a transitive orientation defines a linear order. The following example will illustrate how posets relate to interval graphs. In an attempt to model the non-transitivity of indifference, Luce developed a model for preference motivated by the concept of threshold in psychology, see [32]. He contended that, for a set of things X and the preference relation R, one seek a function f : X -t R and a just-noticeable-tolerance o > 0 with xRy (x is preferred to y) if and only if f(x) > f(y) + 6; i.e., if the value placed on x is sufficiently larger than the value placed on y. This representation led to the development of an order called a semi-order. A semi-order < on X is a binary relation with the following properties: For all x, y, z, w E X, (1) < 10

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is irreflexive, (2) x < y and z < w imply x < w or z < y, (3) x < y and y < z imply x < w or w < z. Scott and Suppes later characterized semi-orders via the existence of a function f and a tolerance 6 as above, see (42]. In his Ph.D. thesis, Roberts studied graphs with adjacency determined by the rule (A): vertices u and v are adjacent if and only if lf(u)-f(v)l::::; 6. Now, observe that a preference relation represented by f gives rise to a transitive digraph, now make this digraph a graph by including a reversed copy of each arc, and finally take the complement of this graph. What you get is a graph whose adjacency is determined by (A), and it turns out, an interval graph in which all intervals have length 6 -the unit interval graphs. We have observed that the complements of unit interval graphs can represent preference relations, whence unit interval graphs are also called indifference graphs, and they are the incomparability graphs of semi-orders. We extend the idea of indifference graphs, without the psychology appli cation, to IkGs. Given 6 > 0, define k functions fi : Xi --t IR (1 ::::; i ::::; k), and a k-partite graph G = (X1 ... Xk, E) such that uv E E if and only if lfi(u)-fi(v)l::::; 6, u E Xi,v E Xj and i i= j. It is easy to see that the graph G is an IkG where all intervals are of length 6: a unit interval k-graph (UikG), or a valuation k-graph, with fi a valuation function In Chapter 5 we show UikGs are cocomparability graphs in 2 ways and hence they are incomparability graphs of some poset of width k. This is because, as we have remarked above, any IkG is k-colorable and hence has k independent sets which in turn form complete graphs in the complement, and a complete graph yields a linear order. The structure of the corresponding posets and of UikGs is an open problem, except 11

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when k = 2, the unit interval bigraphs (UIBGs). UIBGs are precisely the in comparability graphs of a poset of width 2, and equivalent to many other classes of graphs and are characterized by a variety of structural properties. Motivation driving the model for UikGs includes modeling the simultaneous rankings of not necessarily distinct sets, the X/s, by multiple judges, modeled by the f/s. Un der this interpretation, a transitive orientation of a UikG models the preference structure on each partite set, and if the valuation functions are constructed with respect to some uniform scale, then the indifference and preference structure is captured in the graph and complement, respectively. The paragraph preceding this one is intended to suggest potential applicability of UikGs, perhaps in a way similar to unit interval graphs. A unit probe interval graph (UPIG) is a PIG in which all intervals can be made to have the same length in some representation. In Chapter 6 UPIGs are shown to be cocomparability graphs. Thus, by taking the complements of the bipartite graphs, that we conjecture to be a complete list of forbidden subgraphs for bipartite UPIGs, one obtains the forbidden sub-posets that would serve as a forbidden sub-poset characterization for the width-two partial orders that correspond to bipartite PIGs (provided the conjecture is true). The partial orders corresponding to UPIGs generalize semi-orders similar to the way PIGs generalize interval graphs. More specifically, the nonprobes form an independent set by definition, and hence a complete subgraph in the complement Any complete graph has a complete transitive order, that is, a linear order. Hence the partial orders corresponding to UPIGs are an interplay of a semi-order induced on the probes, and a linear order induced on the nonprobes. These orders have 12

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not been studied. Perhaps there is an application in psychology similar to the one for semi-orders and unit interval graphs. 1.4.2 (0, 1)-matrices Any m x n (0, 1)-matrix corresponds to a bipartite graph G = (X, Y, E) with lXI = m and IYI = n; and conversely, with A( G) denoting the (0, 1)-matrix obtained from G with entry (i,j) = 1 if and only if Xi and Yi are adjacent in G. A (0, 1)-matrix has a zero-partition (ZP) if there are independent row and column permutations after which the zeros may be labeled R or C so that to the right of every R is an R and below every C is a C. A (0, 1)-matrix has the consecutive 1 's property for rows (C1R) if the columns can be permuted so that the 1 's in every row appear consecutively; the consecutive 1 's property for columns (C1C) is defined similarly. A (0, 1)-matrix has a monotone consecutive arrangement (MCA) if it has a zero partition with the additional property that below and to the left of every C is a C and above and to the right of every R is an R. Let M be a (0, 1)-matrix; then if M has an MCA, it has C1R and C1C; if M has C1R or C1C, then it has a ZP. These containments are proper and each (0, 1)-matrix property characterizes a class of IBGs. If G is an IBG, then A( G) has a ZP, and conversely (see Chapter 3). Recall that if one of the families of intervals representing an IBG G can be restricted to be points, then we call G an interval-point bigraph. If A(G) has C1C or C1R, then G is an interval point bigraph and conversely (see Chapter 4). If A( G) has an MCA, then G is a UIBG (or belongs to any of the equivalent classes), and conversely (see Chapter 5). In [45], A. Tucker gave what he called a structure theorem for C1C: he determined a minimal forbidden list of bipartite graphs whose corresponding (0, 1)-matrices 13

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do not have C1C. We conjecture such a list for a (0,1)-matrices having C1C or C1R. The forbidden subgraph characterization for UIBGs in Chapter 5 gives a structure theorem for the (0, 1)-matrices with an MCA. IfiBGs are characterized in terms of forbidden substructures, then we would have a structure theorem for the zero partitionable (0,1)-matrices. 1.4.3 Circular Arc Graphs A circular arc graph is the intersection graph of arcs of a circle. We call the circle from which arcs are obtained the host circle. Circular arc graphs have been extensively studied; at the time this thesis is written, a popular scientific internet search engine gives over 50 articles on circular arc graphs. The seminal papers are probably (46] and [44] by Tucker, while a nice introduction and further references and applications can be found in the texts (20] by Golumbic and (40] by Roberts. A 2-clique graph is a graph in which the vertices can be partitioned into two cliques so that all edges not belonging to the two cliques lie between them; a bipartite graph's complement is a 2-clique graph. The 2-clique circular arc graphs are characterized by a list of bipartite graphs that cannot occur in their complements in [36]. In [12] Feder, Hell, and Huang give a different characterization of 2-clique circular arc graphs via a forbidden substructure that captures the prohibitory property in all the graphs derived from [36]. This new characterization probably inspired the connection between interval bigraphs and a restricted class of 2-clique circular graphs proved in [24]. Also in the same work, proper interval bigraphs are shown to be precisely the complements of proper circular arc graphs. In [44] it is shown that proper circular arc graphs are necessarily 2-clique graphs. Capitalizing on this trend, 14

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we prove that interval point bigraphs are identical to the 2-clique circular arc graphs in which the vertices of one of the covering cliques can be labeled in a circularly consecutive fashion. The next two sections give more thorough summaries and introductions to the meatiest chapters. 1.5 Progress on IBGs We discuss the content of Chapter 3. As this thesis is written, there is no forbidden subgraph characterization for interval bigraphs (henceforth IBGs) in general. Although [22] purports to such a characterization, it does not We give a complete characterization for the IBGs that are cycle-free in Theorem 3.3. But there are several ways, other than by forbidden subgraphs, to characterize IBGs at this point. P.Hell and J. Huang showed that IBGs are precisely the complements of 2-clique circular arc graphs in which no two arcs cover the host circle, and are characterized by two different orderings of the vertices, see [24]. In [10] interval digraphs are characterized via the structure of their adjacency matrices, and we do the same for IBGs (in terms of their reduced adjacency matrices). A collection of subgraphs g = { Gi} is consecutively ordered if v E Gin Gk implies v E Gj for any vertex v and i < j < k. We also characterize IBGs via an edge cover of bicliques that can be consecutively ordered, motivated by various consecutive order characterizations for interval-type intersection graphs in the literature. A new construction for a representation of an IBG via its reduced adjacency matrix is given. An asteroidal triple (AT) of a graph G is a set of three vertices such that 15

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there is a path between any pair that avoids the neighborhood of the third. In [4] interval graphs are characterized as those chordal graphs that have no AT in any induced subgraph. An IBG analogue for an AT is an asteroidal triple of edges (ATE): a set of three edges such that there is a path between any two that avoids the neighborhood of the third edge, where if e = uv E E, then N(e) = N(u) U N(v). In Chapter 2 it is shown that an ATE is a forbidden substructure for a graph to be an interval k-graph, and hence an IBG, but the converse is not true. In [37] it is conjectured that G is an IBG if and only if it has no ATE and no insect (a class of graphs defined in Chapter 3), see Figure 3.12. But in [24] this conjecture is refuted, the list of insects is extended, and the modified would-be conjecture is purported to be refuted by undisclosed examples. At the Rocky Mountain Discrete Math Days, P. Hell claimed that the examples he and J. Huang are large. Also in Chapter 3, we summarize what is known, and what can be deduced from the literature, regarding the forbidden subgraph characterization for interval bigraphs. 1.5.1 :F Restricted to Unit Intervals While :F with no restrictions renders the characterization problem for probe interval graphs and interval k-graphs difficult, if :F is restricted to consist of unit length intervals, giving UPIGs and UikGs, the problem becomes more tractable. It turns out that a surprisingly lengthy list of characterizations for unit interval bigraphs can be made when the various results are put together; such a list is developed in Chapter 5. A graph G = (V, E) is a tolerance graph if there is a function t : V -7 lR 16

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and a collection of intervals { Iv}, Iv corresponding to v E V such that uv E E if and only if Jlv n lui min{t(u), t(v)}. G is a unit tolerance graph (UTG) if it is a tolerance graph and Jivl = 1 for each v E V. Any probe interval graph is a tolerance graph. To see this, take a probe interval graph G = (P, N, E), put t(v) = maxvEv{!IvJ}, and t(u) = E, for u E N, where E is an arbitrarily small positive constant. Also any UPIG is a UTG, but not conversely. In Chapter 6 UPIGs are characterized in the cycle-free case and in the bipartite case a characterization is conjectured that is substantiated by their relationship to UTGs. Specifically, the results regarding bipartite UTGs in [3] substantiate the conjecture. UTGs have interesting alternative representations. A parallelogram graph is the intersection graph of parallelograms in which two edges lie on parallel lines. In [2] it is shown that G is a UTG if and only if it is a parallelogram graph. Therefore, UPIGs belong to the class of parallelogram graphs. This fact yields an automatic proof that UPIGs are cocomparability graphs, and suggests that their containment in the class of parallelogram graphs could be further capitalized upon. 1.6 Summary of Chapter Contents In Chapter 2 the stage is set with the introduction of interval k-graphs. They are shown to be a class of perfect graphs, to be a generalization of probe interval graphs, and are characterized by the presence of consecutive subgraphs. Chapter 3 focuses on interval bigraphs. They are characterized via their reduced adjacency matrices and a summary of known characterizations and forbidden substructures is given. In Chapter 5 unit interval bigraphs are characterized in 17

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over a dozen ways, including a forbidden subgraph characterization, a reduced adjacency matrix characterization, and a vertex ordering characterization. Also, following Roberts, unit interval bigraphs are shown to be equivalent to proper in terval bigraphs. A consequence of the characterization for unit interval bigraphs is a structure theorem for the (0,1)-matrices with a monotone consecutive ar rangement. Other consequences are, of course, that a surprisingly large list of other mathematical objects have a myriad of ways to be represented. Interval point bigraphs are characterized via their reduced adjacency matrices, via con secutively orderable subgraphs, and shown to be equivalent to a restricted class of probe interval graphs and a restricted class of circular arc graphs in Chapter 4. We make a step toward a forbidden subgraph characterization for unit probe interval graphs by giving a complete list of forbidden subgraphs for cycle-free UPIGs and conjecture a complete list for unit probe interval graphs in the bipartite case. In the concluding chapter, the open problems are summarized, and future directions for research are discussed. 18

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2. Interval k-graphs and Probe Interval Graphs In this chapter, we introduce our largest class of graphs, interval k-graphs. We give a characterization via consecutively orderable subgraphs, show that they generalize probe interval graphs, and give two proofs that they are perfect. We define and prove a forbidden substructure for interval k-graphs that is analogous to an asteroidal triple, and, because of the containment relationship we show, is also a forbidden substructure for probe interval graphs. We end the chapter with a couple of applications. 2.1 Definitions and a few details A proper coloring of a graph G is an assignment of vertices to colors so that adjacent vertices have different colors. So, with G a graph, and f : V(G) ---+ {1, ... k} the function that assigns a color to each vertex, if vertices u and v are adjacent, then f(u) ::/= f(v). The subset V(G) defined by Vi= {v E V(G): f ( v) = i} is called color class i; by definition Vi is an independent set in G, that is, G(Vi) has no edges. A graph G with a proper coloring is an interval k-graph (IkG) if each vertex v can be assigned an interval I (v) of JR such that vertices are adjacent if and only if their corresponding intervals intersect and each vertex has a different color. The collection of intervals {I ( v) : v E V} together with the coloring is an interval k-representation for G. We say Ii = {I(v) : v E Vi} is an interval class of the representation and that both v and I ( v) belong to interval class i. We may consider the collection of intervals in a representation as a family F = {I1 ... ,Ik}, where Ii denotes interval class i, fori= 1, 2, ... k 19

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So, an IkG can be thought of as the intersection graph of a family of intervals of R in k classes with vertices adjacent if and only if their intervals intersect and belong to different classes. Recall that a graph r is perfect if for every induced subgraph H of r, x(H) = w(H), where x(G) is thefewest number of colors needed to properly color G and w(G) is the number of vertices in the largest induced clique of G. Not every proper coloring of G is conducive to an interval k-representation. For example, if the vertices of a 4-cycle are each assigned to a different interval class, then no interval k-representation is possible. The reasoning is the same as that forbidding a 4-cycle from being an interval graph. But because IkGs are perfect, which we prove in this chapter, if G is an interval k-graph, then in order to give an interval k-representation for G, we need only w(G) interval classes; more than w (G) interval classes will not allow for a representation when w (G) does not. A probe interval graph (PIG) is a graph G = (V, E) in which V can be partitioned into subsets P and N called probes and nonprobes, respectively, and to each v E V there corresponds a unique interval I(v) C R with uv E E if and only if I(u) n I(v) # 0 and at least one of u, v belongs toP. For G = (V, E) a PIG, the collection {I ( v) : v E V} together with the vertex partition is called a probe interval representation. Note that G(N) is an independent set, G(P) is an interval graph, and any interval graph is a probe interval graph with N = 0. These observations do not characterize probe interval graphs, but they do characterize a related class of graphs. A graph G = (V, E) is an interval split graph if V can be partitioned V = (U1 U2 ) so that G(U1 ) is an interval graph and U2 is an independent set. Clearly, every PIG is an interval split 20

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graph, but the converse is not true; see discussion regarding Figure 2.1. Also, if a given partition V = (P, N) is forced, it may prohibit a graph from having a probe interval representation, even though the graph may have a probe interval representation using another partition For example in any induced 4-cycle of a PIG, exactly one of the pairs of non-adjacent vertices must be contained in N. Hence, if in any induced 4-cycle, adjacent vertices are forced to belong to P, then the graph is not a PIG. If G = (V, E) is a PIG with partition V = (P, N) forced, then we say that G is a partitioned probe interval graph. Proposition 2.1 If G is a probe interval graph, then in any induced 4 cycle exactly one of the pairs of non-adjacent vertices must belong to N. Proof: Let G = (V, E) be a PIG with C = {1, 2, 3, 4} c V and (1, 2, 3, 4, 1) an induced 4-cycle. Clearly no three vertices of C can belong to N and no ad jacent vertices can both belong to N. Assume that only one vertex belongs to N; relabel if necessary, and let 1, 2, 3 E P, and 4 E N. !(1) n 1(2) =/= f/J, J(2) n !(3) =/= (/) and 1(1), 1(3) must be disjoint. So, we may assume r(1) < l(3). But since 1(4) must intersect both J(l) and 1(3), it intersects 1(2), forcing 4 ++ 2 contradiction. Thus, the only possibility is for 1, 3 E N or 2, 4 E N. We assume 2, 4 E N and give a probe interval representation for G(C). Put J(1) = [1, 3], 1(3) = [4, 6], and 1(2) = 1(4) = [2, 5]. The result follows because any induced subgraph of a PIG is a PIG. Theorem 2.9 and Theorem 2.7 will show that a 4-cycle is the largest cycle a PIG can have as an induced subgraph. Now we put Proposition 2 1 to work and give an example of an IkG and an 21

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e f !;..: __ ci::: =='=== d====== a==::::i:::::::='::J Figure 2.1: An IkG that is not a PIG interval split graph that is not a PIG. M2 of Figure 2.1 is an IkG, and an interval split graph, but not a PIG. Beside M 2 is an interval k-representation, so M2 is an IkG. Put U1 = {b, c, e, f} and U2 ={a, d}; M2(U1 ) is a path on 4 vertices, which is an interval graph, U2 is an independent set, so M2 is an interval split graph. We prove that M2 is not a PIG using Proposition 2.1. By way of contradiction, assume that M2 is a PIG and consider {a, b, c, d}. By Proposition 2.1 either a, d E N or c, b E N. In the former case d, f E N is forced since {a, b, c, f} induces a 4-cycle, but then adjacent vertices d and f are in N; similarly, in the latter case b and e are forced to be non probes, because {a, c, d, e} induces a 4-cycle. We have adjacent nonprobes in either case, contradicting the fact that nonprobes form an independent set in a PIG. 2.2 A characterization for IkGs and a containment relationship A set of distinct induced subgraphs g = { G1 ... Gt} of a graph G = (V, E) is consecutively ordered when for each v E V, if i < j < l and v E Gin G1, then v E Gj. We will say that g covers G if it forms an edge cover of G; i.e., if every 22

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edge belongs to some H E Q. Theorem 2.2 supports Theorem 2.3 and both give context to Theorem 2.6. Theorem 2.2 (Fulkerson, Gross, 1965, [13]) A graph is an interval graph if and only if its maximal cliques can be consecutively ordered. A quasi clique Q in a PIG G = (P, N, E) is a set of vertices with all vertices of Q n P adjacent, and any vertex of Q n N adjacent to all vertices of Q n P. A maximal quasi clique of G is a quasi clique that is not contained in any larger quasi clique. A complete set of maximal quasi cliques of G is a collection of maximal quasi cliques in which each maximal clique of G is in exactly one maximal quasi clique of the set. A collection of sets is said to have the Helly property if whenever a subcollection sl, ... 'sk of them intersect pairwise, then Si is nonempty. Any collection of intervals has the Helly property. Theorem 2.3 is purportedly a consequence of Theorem 2.2 and because intervals have the Helly property, see [35]. Aside from this claim, there is no published proof of Theorem 2.3, so we attempted to give one here, but the result we proved turned out to be a stronger result; it is Theorem 2.4. Theorem 2.3 (Zhang, 1994, (48]) An interval split graph G = (U1 U2, E), U2 an independent set, is a probe interval graph with respect to the same partition U1 = P, U2 = N if and only if there is a complete set of maximal quasi cliques that can be consecutively ordered. Theorem 2.4 Let G = (U1 U2 E) be a graph with G(U2 ) an independent set. G is a PIG with P = U1 N = U2 if and only if G has an edge cover of quasi cliques that can be consecutively ordered. 23

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Proof: Let G = (U1 U2 E) be a graph with G(U2 ) an independent set. Suppose Q = {Q1 ... Qs} is an edge cover of quasi cliques that is consec utively ordered. For each v E V define an interval I(v) = [l(v), r(v)], where l(v) = min{i: v E Qi}, and r(v) = max{j: v E Qj}, and N = U2,P = U1 We claim {I ( v) : v E V} together with the partition is a probe interval repre sentation for G. By definition of Q, for vertices u, v, we have I(u) n I(v) ::J 0 if and only if they belong to the same quasi clique, in which case uv E E unless u, v E N = U2; but if u, v E N, then the intersection of their intervals does not induce an edge in G. Thus, G is a probe interval graph with U1 = P and U2=N. Now suppose G has a probe interval representation I= {I(v)}vEV, and let r1 < r2 < < r m be the distinct right endpoints among intervals of I. Define Qi to be the subgraph of G induced on U{v: r;EI(v)} v; this subgraph is a quasi clique, and the collection Q = { is a consecutively ordered collection of quasi cliques that covers the edges of G. We will use the next result regarding where maximal cliques of a PIG are in a consecutively ordered cover of quasi cliques to prove a necessary condition for bipartite PIGs in Chapter 4. Lemma 2.5 If G is a PIG, then the consecutive cover of quasi cliques Q can be made so that each maximal clique of G is contained in exactly one quasi clique ofQ. Proof: Let G be a PIG with Q a consecutively ordered edge cover of quasi 24

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cliques defined by the distinct right endpoints of the probe interval representation for G as in the proof of Theorem 2.4; that is, Qi = G ( U{v: r;EI(v)} v), where r1 < < r m are the distinct right endpoints. A maximal clique C of G is a quasi clique containing at most one nonprobe. No C is contained in more than one Qi because each vertex v with interval [l ( v), ri] will not be contained in Qi+l Now we show that every Cis contained in at least one Qi. Assume C consists of Po C P and n E N. We must have nP: pEPo I(p) n I(n) containing some common point q by the Helly property. Either q = ri or q E (ri, ri+1), for some i. If q = ri, then Cis contained In Qi. If q E (ri, ri+I), then Cis contained in Qi+l, since r(v) 2: ri+l for all v E C. We have shown that Cis in at least one and no more than one Qi, for some i. The next result has a flavor similar to that of Theorem 2.4 and Theorem 2.2 and will serve in showing that any probe interval graph is an interval k-graph. Theorem 2.6 A graph is an interval k-graph if and only if there exists a cover of completer-partite subgraphs that can be consecutively ordered, where 1 :::; r :::; k for each subgraph. Proof: Suppose G = (V, E). is an interval k-graph with representation I = {I(v) = [l(v), r(v)] : v E V}. Index the vertices v1 v2 ... Vn so that r(vi)::; r(vi) whenever i < j. For each i E {1, ,n}, the collection of vertices whose intervals contain r(vi) induces a completer-partite subgraph Gi of G, for n 1 ::; r ::; k. Define Q = U Gi; then Q is certainly a cover of G, and the indexing i=l is a consecutive ordering since any collection of intervals has the Helly property. 25

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Now suppose that Q = { Gb ... Gm} is a cover of consecutively ordered r-partite subgraphs. Assign the vertices of each partite set of G1 to a distinct interval class and start by letting each interval representing a vertex of G1 be a point at 1 E JR. For integers i = 2, ... m, and for each vertex in Gi-l n Gi, extend its interval from i -1 to i E R Assign the vertices in each partite set of Gi \ (Gi-l n Gi) to their corresponding interval classes. This can be done unambiguously because a vertex in Gi must be in the same interval class as any non-adjacent vertex in Gi, and vertices belong to distinct interval classes when ever they are adjacent in G i Since each Gi E Q is an induced subgraph, the interval class assignments will not change for vertices contained in two or more consecutive subgraphs. Therefore, we have an interval k-representation for G. Now we can easily show that the class of probe interval graphs is contained in the class of interval k-graphs. Theorem 2. 7 Every k-chromatic probe interval graph is an interval k-graph. Proof: Suppose G = (P, N, E) is a k-chromatic PIG; then G has a cover of maximal quasi cliques Q that can be consecutively ordered. Each maximal quasi clique Q E Q is a K1 ... 1, t, where t is the number of nonprobes in Q and P n Q is a clique, in other words, a K1 ... 1 Thus, Q is a cover of complete multipartite subgraphs that is consecutively ordered. Figure 2.2 is an illustration of Theorem 2.6. The four distinct right endpoints indicated by the vertical dotted lines induce, in the consecutive order, a K1,2, 26

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9 5 8 6cJ===I 2 ... vj :R, ,v 5 Figure 2.2: An IkG and its consecutive multipartite subgraphs a K1,1 ,2 a K2 2 and a K 1 1 ,2 on the sets {1, 2, 4}, {1, 3, 4, 5}, { 4, 6, 7, 8}, and {6, 7, 8, 9}, respectively Although every PIG is an interval split graph, and every PIG is an IkG, the containment relationship between interval split graphs and IkGs is not known. In the next section we prove IkGs are weakly chordal (and hence contain no induced cycle of length greater than or equal to 5), but a cycle of any length is an interval split graph. Thus, there are interval split graphs that are not IkGs, but the converse is not known -but has not received much attention. We summarize the known containment relationships addressed here by: interval graphs PIGs IkGs with a 4-cycle and M2 serving as separating examples 2.3 Some properties of IkGs 27

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Recall that a graph G is perfect if x(H) = w(H) for every induced subgraph H of G. Let ik (G) be the fewest number of interval classes needed in an interval k-representation of G, and define ik( G) = oo in case G is not an interval k-graph. The next result shows that ik( G) = w( G) provided that G has an interval krepresentation. Note: in any proper coloring, the vertices in a clique each require a different color, hence, w(G) :::; x(G), for any graph G. Given G an IkG, the next result shows how to color G with w(G) colors, showing x(G) = w(G). Since IkGs have the property that any induced subgraph of an IkG is an IkG, this is called the hereditary property, IkGs are perfect. The result also shows that the number of interval classes needed to give an interval k-representation for a graph if it is an interval k-graph is the size of the largest clique. Theorem 2.8 Interval k-graphs are perfect. Proof: Let G be ann-vertex IkG with interval k-representation I= {I(v) : v E V(G)}, where the vertices have been colored with {1, 2, ... r }, r :::; n. Sup pose !(vi) = [li, ri], and relabel the vertices so that li :::; lj if i < j. Sequen tially color the vertices in order v1 v2 ... Vn, recoloring if necessary, using the first available color. We claim that this algorithm produces an w( G)-coloring. Clearly v1 can get color 1. Assume that at step i we need color k, that is, vi needs color k, and that up to this point we have used no more than w( G) colors. Since vi requires color k, it must be adjacent to vii, vi2 ; vik-I colored with 1, 2, ... k -1, where i 1 < i 2 < < ik-l < i. Thus, rij 2 li for each j satisfying 1 :::; j :::; k -1, and so G ( {vii, Vi2 vik-I vi}) is a clique on k vertices, in other words, w(G) 2 k. By induction, this procedure will produce an w(G)-coloring. Therefore, x( G) = w( G) for G an interval k-graph. 28

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A graph G is weakly chordal if neither G nor G contains an induced cycle of length greater than 4. The next result shows IkGs are weakly chordal (and hence so are PIGs) and provides us with examples of forbidden IkGs: the complement of a 6-cycle in Figure 2.3, and any cycle of length 5 or greater. Theorem 2.9 Interval k-graphs are weakly chordal. Proof: Assume C = (v1 v2 vn) is a cycle of length n 2: 5 in an IkG, G. Assume that J( v1 ) is a leftmost interva,l in the representation for C and without loss of generality that r(v1 ) = 1 and J(v1 ) E I1 Then l(v2 ) ::; 1 and l(vn) ::; 1, and since v2 Vn, they belong to the same interval class, say I2 I(v3 ) must belong to an interval class distinct from I2 and (J(v2 ) n J(v3)) n I(vn) = 0 necessarily, so r(v2 ) 2: l(v3 ) 2: r(vn) 2: 1. I(vn_1 ) must belong to a class distinct from I2 and 1::; l(vn-l) ::; r(vn) implies that l(vn-1 ) E J(vz), soC has a chord. Suppose now that G contains Cn, the complement of a chordless cycle of length n. If n = 5, then there is a 5-cycle in G, which contradicts what we have just proved. For n = 6, we prove that the graph in Figure 2.3, which is the complement of a 6-cycle is not an IkG. We use the labels in Figure 2.3 for the proof. By symmetry, we may assume that 1(2) n !(4) lies between 1(1) n !(6) and 1(3) nJ(5). One of 1(2), !(3) belongs to a class different from that to which 1(6) belongs; so no point in ni=1 2 3J(i) can lie to the left of r(6). If 1(5) n 1(6) extends to the left of ni=1 2 3J(i), then 5 f-7 1 or 5 f-7 2, while if !(5) n !(6) extends to the right of ni=l,2 3J(i), then 6 f-7 2 or 6 f-7 3. We make the observation that there are only two distinct ways to decompose 29

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2 4 3 5 Figure 2.3: The complement of a 6-cycle, which is not an IkG a 4-cycle into consecutively orderable complete multipartite subgraphs: as a K2 2 .. or as two copies of K1 2 In either case, at least one pair of nonadjacent vertices must belong to the same interval class, and these intervals must intersect. For n > 6, consider chordless 4-cycles in Cn = (1, 2, ... n). We must make interval class assignments for the vertices of C;, based on their adjacencies. We may assume for the cycle (1, 4, 2, 5) that vertices 1 and 2 belong to the same interval class. The cycle (2, 5, 3, 6) must contain a nonadjacent pair of vertices whose intervals belong to the same class This pair cannot be 2,3 since 3 and 1 are adjacent in Cn, so 5 and 6 belong to the same class. In the cycle (2, 6, 3, 7, 2), 6 and 7 cannot belong to the same interval class, since 7 f-7 5. Therefore Cn does not have an interval k-representation. In [23] Hayward showed that weakly chordal graphs are perfect and so we have another proof of Theorem 2.8. 2.4 Forbidden substructures An asteroidal triple of a graph G is a set of three vertices with the property that between any two there is a path in G that avoids the neighborhood of the third vertex. In [4] it is proved that a chordal graph is an interval graph if and 30

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Figure 2.4: Examples of ATEs only if it does not contain an asteroidal triple in any induced subgraph. There is a forbidden substructure for IkGs that is, in principle, similar to an asteroidal triple. Define the neighborhood of an edge to be the union of the neighborhoods of its vertices. An asteroidal triple of edges (ATE) in G is a set of three edges such that for any two there exists a path in G that avoids the neighborhood of the third edge. Figure 2.4 has two bipartite graphs each with an ATE. Theorem 2.10 If a graph G has an asteroidal triple of edges then G is not an interval k-graph. Proof: Let G = (V, E) be an IkG and x 1 x 2 y 1 y 2 z 1 z 2 E E an ATE of G. We use the notation I(uv) = I(u) n I(v), where u, v E V and uv E E. If I(x1x 2 ) n J(y1 y 2 ) =I 0, then each of N(y1), N(y2 ) contains at least one of x 1 x 2 and no path from z 1 z 2 to x 1 x 2 can avoid the neighborhood of Y1Y2 Therefore I(x1x2), I(YlY2), and I(z1z2) are disjoint, and we may assume that I(y1y2) lies between the other two. Let P = (p 1 ... ,Pm) be a path between x1x2 and Z1Z2. Then J(y1y2) n (J(pt) Uf(p2) U UJ(pm)) C !(pi) for some Pi E P. At least one of J(y1 ), J(y2 ) belongs to an interval class different from that containing !(pi), 31

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so Pi E N(y1 ) U N(y2), contrary to assumption This gives an interesting corollary regarding PIGs. Corollary 2.11 Probe interval graphs are ATE-free. The converse of Theorem 2.10 is not true unless k = 2 and we consider only trees, see Chapter 3. In the proof of Theorem 2.9 it is shown that the ATE-free graph in Figure 2.3 is not an IkG. We present two graphs that are minimal forbidden subgraphs for IkGs, one with an ATE and the other ATE-free. Refer to Figure 2.5. H2 is ATE-free and we use Theorem 2.6 to show that it is not an IkG, while the thick edges of H1 are an ATE. 9 2 10 Figure 2.5: Two minimal forbidden graphs for IkGs Consider H2 -5; we may take the subgraphs induced on {1, 3, 4}, {2, 3, 4}, {3, 4, 6, 8}, {3, 6, 8, 10}, {7, 8, 10}, {8, 9, 10} in that order as our consecutive complete multipartite subgraphs. For H2 notice that the clique on {3, 6, 7} must be contained in both 4-cliques, which necessitates taking the 4-cliques consecutively. Three of the 5 pendant vertices must be taken in subgraphs either 32

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before or after the 4-cliques, and it is evident that 3 subgraphs containing a triple of pendants cannot be ordered without taking a 4-clique within that ordering. Both graphs of Figure 2.5 belong to families of minimal forbidden subgraphs for IkGs. A forbidden subgraph characterization for interval k-graphs seems rather difficult at this time. However, we posit the following conjecture. We require that G have x( G) = 3 because in Chapter 3, we will see that the following is true: For G bipartite and chordal, G is an interval bigraph if and only if it has no ATE. Also H2 of Figure 2.5 is 4-chromatic, chordal and has no ATE. Conjecture 2.12 Let G be a chordal graph with x(G) = 3 and no ATE. Then G is an interval 3-graph. 2.5 Applications of IkGs Let C be a set of chemicals or compounds, each with temperature constraints for its storage. Suppose subsets of the chemicals react with one another or pose undesired contamination threats when combined, and therefore cannot be stored together. Break C into { C1 ... Ck}, where each Ci consists of chem icals or compounds that cannot be stored together regardless of temperature compatibility. Furthermore let C = { Cij }, where the first subscript indicates to which Ci the compound or chemical cij belongs and the second subscript enu merates the members of Ci, so 1 j \Gil The temperature constraint for each Cij will be a closed interval of IR: t(Cii) = [l(ij), u(ij)]. Model the storage of C by the graph G = (C, E), where CijCrs E E if and only if t(Cij) n t(crs) =/= 0 and i =/= r. In other words, G is a graph with C as its vertices and vertices are adjacent if they may be stored at the same temperature and they pose no threat 33

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of reaction or contamination when stored together. Observe that a clique in G corresponds to a group of chemicals that can be stored together and at the same temperature. Therefore, the minimum number of temperature regulated containers required to store Cis precisely the minimum number of cliques which cover the vertices of G. Now consider interval 2-graphs; i.e., interval bigraphs. A typical application of a bipartite graph G = (W, J, E) is to have one partite set W = { w} correspond to workers and the other J = {j} to jobs with wj E E if w can do j. We modify this application to include constraints on when the workers work and when the jobs can be done. We assume that the ability of a worker to do a particular job rests on availability only; that is, if worker w is employed while job j can be done, then w can do j. Here we assign intervals to each vertex of G = (W, J, E): I(v) = [b(v), e(v)], corresponding to the times the worker's shift begins and ends or when the jobs availability begins and ends. So, wj E E if and only if I( w) n J(j) =/= 0, and an optimal assignment of workers to jobs corresponds to a maximum matching in G; as in the standard application. 34

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3. Interval Bigraphs Our attention in this chapter is restricted to interval 2-graphs, or interval bigraphs. We give a characterization of them via their reduced adjacency matrices, present the characterizations in the current literature, and summarize the progress on the forbidden induced subgraph characterization. 3.1 Background and definitions An interval bigraph (IBG) is the intersection graph of two distinct families of intervals with vertices adjacent if and only if their corresponding intervals overlap, and the intervals belong to different families. In other words, interval bigraphs are interval 2-graphs. The two families of intervals that represent an IBG will be called the interval bigraph representation, and with G = (V, E) an IBG, I(x), x E V denotes the interval corresponding to vertex x, while l(x) and r(x) denote the left and right endpoints of I(x), respectively. Figure 3.1 shows that HlO is an IBG, since to the right of it is an interval bigraph representation. a j h c d e f i-g= J= f= a-c h-e-b= d==== Figure 3.1: HlO and an interval bigraph representation. Recall that a probe interval graph can be defined as follows. A graph G = (V, E) is a probe interval graph if there is a partition of V = (P, N) and an 35

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interval can be assigned to each vertex so that vertices are adjacent if and only if their corresponding intervals overlap and at least one of the vertices belongs to P. So, if at 'at least' is replaced by 'exactly', then we have another definition for IBGs. This natural extension of the notion of a probe interval graph is what led us to study IBGs. Because of the application probe interval graphs model, we originally called IBGs "probe-clone bipartite interval graphs". But this unwieldy name bogged us down and so they became interval bigraphs. This nomenclature allowed us to find pre-existing research by Harary, Kabell and McMorris, [22], and H. Miiller, [37], which makes our research more relevant In this chapter, we will characterize IBGs by the structure of their reduced adjacency matrices, and start a thread that we continue for the rest of this thesis: we give matrix characterizations for interval point bigraphs and unit interval bigraphs in the chapters that follow. As a corollary to Theorem 2.6 IBGs are characterized by the existence of a consecutively orderable edge cover of bicliques. Recent work by P. Hell and J. Huang, [24], identifies the complements of IBGs with a class of restricted 2-clique circular arc graphs. The (0, I)-matrix property that will turn out to describe precisely the structure of the reduced adjacency matrix of an IBG is as follows. This property was introduced in [IO] to characterize interval digraphs. We use properties of bipartite graphs to get Theorem 3.4. A matrix with a zero-partition (ZP) is a (0, I)-matrix with independent row and column permutations after which each zero can be labeled R or C so that every entry below a Cis a C and every entry to the right of an R is an R. Figure 3.2 shows an example in which rows 6 and 7 are swapped and then columns 6 and 7 are placed between columns I and 2 36

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1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 0 0 1 1 (_____ 0 0 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 l 1 1 1 l 1 1 1 l l 1 1 1 1 11RRRRR 1111RRR C1111RR C11C11R C11CC11 C 11CCCR CC 1CCC 1 1 1 1 1 l (_____ 1 1 1 Figure 3.2: A zero partitionable matrix 1 1 l 1 1 l 1 1 l 1 1 1 1 1 to obtain a zero partition. We will show that IBGs are precisely those bipartite graphs whose reduced adjacency matrices are zero-partitionable. A more restricted property for (0, I)-matrices is the consecutive l's property. A (0, 1)-matrix has the consecutive 1 1S property for rows if the columns can be permuted so that the l's in every row appear consecutively; the consecutive 1 1S property for columns is defined similarly. Tucker investigated the structure of (0, I)-matrices with the consecutive l s property for columns by determining which bipartite graphs do not have the consecutive l's property for columns in their reduced adjacency matrices. He dubbed his result a "structure theorem for the consecutive l's property", see [45]. A bipartite graph G = (X, Y, E) is X -consecutive if the vertices in X can be ordered so that for each y E Y, N(y) is a consecutive set in X with respect to the ordering found for X. Note: 37

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in this thesis rows of the reduced adjacency matrix are always indexed with X, and columns with Y, when representing a bipartite graph with a matrix; conversely, rows will correspond to X when representing the matrix as a graph. Clearly with this convention, an X -consecutive bipartite graph has a reduced adjacency matrix with the consecutive l s property for columns. The following result of Tucker s will be used in this chapter and the next. It shows that a structural characterization for the consecutive 1 's property for columns is tantamount to determining all bipartite graphs G = (X, Y, E) with an asteroidal triple contained in X, where the bipartition is predetermined. Theorem 3.1 (Tucker, 1972, [45]) A bipartite graph G = (X, Y, E) is Xconsecutive if and only if there is no asteroidal triple of G contained in X; i.e., A( G) has the consecutive 1 's property for columns if and only if G(A) has no asteroidal triple contained in X. We will use Theorem 3.1 to show that having a zero partition is a property less restrictive than having the consecutive l's property for rows or for columns. To this end we present the following proposition which shows that the class of zero partitionable matrices properly contains those with consecutive l's in rows or in columns Note that we use Theorem 3.1 to show that the consecutive property for columns is not present by reversing the roles of X and Y. Proposition 3.2 If a (0, !)-matrix has the consecutive 1 's property for rows or for columns, then it has a zero partition, but not conversely. Proof: Let M be an m x n(0, 1)-matrix with the consecutive 1's property for rows (columns). Let ai (/3j) be the first column (row) in which a 1 appears 38

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in row i (column j), 1 :s; i :s; m (1 :s; j :s; n). Permute rows (columns) so that { ai} (j3i) forms a nondecreasing sequence. Now, label all zeros to the right of (below) the last 1 in each row (column) with an R (a C). The matrix that rffiill:::::::: aim zero partition: [i: but it is there-oo11o cc11R 00101 CC1C1 duced adjacency matrix of H10 in Figure 3.1, which has an asteroidal triple on each of the sets {a, i, e }, {b, j, f} and each one is contained in a distinct partite set of H10. Therefore, by Theorem 3.1, A(H10) separates the classes of zero partitionable matrices from that of (0, 1 )-matrices with consecutive 1 's in rows or in columns. 3.2 Characterizations for IBGs First we completely characterize the trees that are IBGs. Note that IBGs have the hereditary property, that is, any induced subgraph of an IBG is an IBG, so if an induced subgraph of G is not an IBG, then G is not an IBG. We use (vo, v1, v2, ... vn) to deiwte a path of length n, and I(H) denotes the interval representation for the subgraph H. For v a vertex, the size of v' s neighborhood N(v) is JN(v)l = deg(v). Theorem 3.3 A cycle-free graph G is an IBG if and only if it has no NL1 0 of Figure 3. 3 as an induced subgraph. Proof: First, we show that NL10 is not an IBG. Suppose, on the contrary that it is and construct a longest path. Without loss of generality, and using the labeling in Figure 3.3, suppose we have the interval bigraph representation 39

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NLlO j h abc defg Figure 3.3: NL10: The smallest tree that is not an IBG and the only forbidden subgraph for cycle-free IBGs for P = (a, b, c, d, e, !). Because i is not adjacent to c or e, and the length of P is sufficiently large, I ( i) c I (d). Hence I (j) is forced to intersect I (d), contradiction. Now, suppose G is a cycle-free graph with NL10 not an induced subgraph. Let P = (x0 x1 ... Xn be a path of maximum length. Since P maximum length deg(x0 ) = 1 = deg(xn), and deg(u) = 1, for any u E (N(x1)\)P U (N(xn-l) \ P). Since G has no induced NL10 deg(v) = 1, for any vertex v at distance 2 from any vertex of P. Create I(P) with !(xi) = [i, i + 1], for i = 0, ... n and place deg(x1 ) -2 equidistantly spaced points in (1, 2) for its neighbors, and similarly deg(xn-d2 points in (n-1, n) for Xn-1's neighbors. Now for each Xi, 2 ::; i ::; n-2, with deg(xi) > 2, put deg(xi) -2 equidistantly spaced points in (i, i + 1) for xi's neighbors at distance 1 from P, and for any vertex at distance 2 from xi not on P, make its interval a point equal to its (unique) neighbor's point. G is cycle-free, and hence bipartite, so the natural bipartition of G defines the interval classes and the construction renders G an IBG. Now for the matrix characterization for IBGs in general. A standard zeropartition is a zero-partition in which an ambiguous 0 is lal;>eled C if it lies on or 40

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below the last full diagonal, while an ambiguous 0 above this diagonal is labeled R. In symbols, entry (i, j) E D' = { (1, n-(m -1)), (2, n-(m-2)), ... (m, n)} or below D' is labeled C if it can be labeled either R or C, and labeled R if it lies above D' and can be labeled either R or C. We make the following assumptions at the outset: a (0, 1)-matrix has m rows and n columns, m ::; n (since the class of zero-partitionable matrices is closed under taking the transpose), any zero-partitionable matrix will be in standard form. The following example shows how assuming that a zero partition be in standard form precludes ambiguity, since some of the entries in the lower right can be labeled R or C. A standard zero-partition for the matrix A is given below: 11111111 10000000 A= 10000110 f--t 10000100 10000000 11111111 1CCCCRRR 1CCCC11R 1CCCC1CR 1CCCCCCC Theorem 3.4 G is an interval bigraph if and only if its reduced adjacency matrix has a zero-partition. Proof: ( =?) Let G = (X, Y, E) be an interval bigraph with I { Ix,Iy}xEX,yEY as an interval representation with Ix = [l(xi), r(xi)] for i 1, ... lXI = m, and Iy = [l(yj), r(yj)] for j = 1, ... IYI = n. Index XU Y so that l(x1 ) ::; l(x2 ) ::; ::; l(xm) and l(y1 ) ::; ::; l(yn) Form A(G), the reduced adjacency matrix of G with row i corresponding to X i and column j corresponding to Yi. This ordering of rows and columns will enable us to label the zeros of A( G) in accord with a zero-partition. Assume entry (i, j) = 0. Then either r(xi) < l(yi) or r(yi) < l(xi) but not both since l(xi) ::; r(xi) and l(yi) ::; r(yi) If r(xi) < l(yj), then label entry (i, j) 41

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with R. Every entry to the right will be a zero which we may label R. To see this suppose (i,j + s) = 1, for s 1. Then r(xi) l(Yi+s), but l(Yi+s) l(yj) and this implies r(xi) l(yj); contradiction. If (i, j) = 0 and r(yj) < l(xi), then label entry (i,j) with a C. Now suppose (i + s,j) = 1 again with s 1, then r(yj) l(xi+s) and since l(xi+s) l(xi) we have r(yj) l(xi); contradiction. So every entry below (i,j) is a zero we may label C. This shows that A(G) has a zero-partition. ( <=) We create an interval representation for the bipartite graph, G = (X, Y, E), obtained from A = (ai,j),. a zero-partitionable matrix in standard form, in which xy E E if and only if ai,j = 1. We assume m = lXI ::; IYI = n, arid that there are no all-zero rows or columns, since the intervals for isolated vertices may always be placed with no trouble. All intervals will be open subsets of the segment m + 1). Open intervals will serve to preclude singleton point intersections. The standard partition serves to relieve problems that would otherwise arise from sparseness in the lower right of the matrix, and of course to eliminate ambiguity in the labeling. Essentially, { Iy }yEY is created so that Ix may be inserted so that Ixnfy = 0 whenever (i,j) = 0, and Ixnfy # 0 whenever ( i, j) = 1. We define the following symbols which allow us to use the information implicit in the zero-partition: = {first column in which a C appears in row i 0 if there is no C in row i = {last column in which a C appears in row i 0 if there is no C in row i = {first column in which an R appears in row i n + 1 if noR appears in row i We define the analogous indicators for columns: 42

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' = { the last row in which a R appears in column j PJ 0 if there is noR in column j = { the first row in which a C appears in column j fJ m + 1 if there is no C in column j First create { IyJr=1 by specifying left and right endpoints of the interval IYi cor responding toy{ Iyi = (l(yj),r(yj)). Define Sj = j{(i,j): ai.j = R, 1 i < pj}j; note that Sj-1 = Sj when N(yj) and N(Yi-I) agree on x1 x2 ... xP'. J Initialize r(y0 ) = 1, and l(Yn+l) = m + 1. For j E {1, ... ,n}; If p'. = p'. 1. J J-' Put l(yj) = pj + + r(yj) = ')'j; Else put l(yj) = pj + r(yj) = tJ Now we create by specifying left and right endpoints for each interval Ixi corresponding to xi. For i E { 1, ... m}; Put l(xi) = max{r(yj) : cPi j {i, ai,j = C}, r(xi) = min{l(yj) : P i j n+ 1}. Claim 1 The sequence { s j} is nondecreasing. Choose an arbitrary row i and column j, and suppose Sj+l < Sj Then above row i we have at least one or fewer R's in column j + 1 as we have in column j. This implies we have a submatrix of the form [ where E {1, C}. Therefore Sj Sj+l Claim 2 For each i l(xi) i and r(xi) ;:::: i + First we show l(xi) i. If cPi = {i = 0, then l(xi) r(yo) 1 < 43

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for all i. Otherwise we have 1 i; so r(xi) ?: i + Otherwise we have 1 Pi j n, and for each j, l(yj) ?: pj + ?: i + since ai, j = R when j satisfies Pi j n. Claim 3 IIYi I > for all j, and IIx; I ?: By design, IIYi I ?: and IIx; I ?: by Claim 2. Claim 4 The sequence {pj} is nondecreasing. Assume j < k and pj > Then we have a submatrix of the form [ or [ Therefore, pj whenever j < k. Claim 5 The sequence { ri} is nondecreasing. Assume i < j and {i > {j. Then we have a submatrix of the form [ or [ So {i {j whenever i < j. Claim 6 The value min{l(yj): j n+1} = l(yPJ, and hence r(xi) = l(ypJ If Pi= n+ 1, then r(xi) = l(Yn+l) = m+ 1. Fix i arbitrarily and choose k so that k n. Assume = then l(yp;) = = l(yk) since { Sj} is nondecreasing by Claim 1. If =/= then < by Claim 4, and so l(yp;) = + pj + t(Yk) We will now prove that the algorithm is correct by verifying Ixi n Iyi =/= 0 if and only if ai,j = 1. We consider several cases each for [ai,j = 0 ===} Ix;niYi = 0] and [(i,j) = 1 ===} Ix; n IYi =/= 0], which span Claim 7 and Claim 8. Claim 7 If a i,j = 0, then Ixi n Iyj = 0. We deal with ai,j = C and ai,j = R separately. 44

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Case 1 Assume ai,j = 0 and is labeled C. Then r(yj) :s; i and l(yj) :s; iBut l(xi) 2: i since r(yj) :s; i and so l(yj) < r(yj) :s; l(xi) < r(xi) Therefore, Ix; n Iyi = 0, since all intervals are open. Case 2 Assume ai,j = 0 and is labeled R Then r(xi) = min{l(yk) : i; and pj < i otherwise we would h [ (i,'Yi) J [c] 8 1 < l( ) < 1 1 i-1 d ( ) > 1 ave (pj,'Yi) = R o 2 Yi '/.+ 2 + 2m+l < '/. an r Yi 2 + while l(xi) :s; i and r(xi) 2: i + by Claim 2. Therefore, r(xi) > l(yj) and l(xi) < r(yj), which implies Ix; n IYi =I= 0. Case 2 ( i; and pj < i otherwise [ (i, c/Ji) (i, 'Yi) ] [c c] c 1 8 l( ) ( ) > 1 z( ) < d (pj,c/Ji) (pj,'Yi) = R as m ase o Yi < 2, r Yi 2+ Xi 2, an r(xi) 2: i + Therefore, Ix; n Iyi =/= 0. Case 3 (ai,j = 1 and "/i < j < Pi :s; n + 1) As in Case 1 and in Case 2, 'Yj > i; but pj < i is not forced. However, if pj < i, then we may argue as in Case 2. Hence we will assume pj > i, which gives the following set-up: j k i c ... l .. R pj R R where Pi :s; k :::; n + 1. Let pj i + p for p 2: 1. The algorithm gives i + p + :s; l(yj) :s; i + p + + Since l(xi) :s; i from Claim 2, we have l(xi) < l(yj) Assume pj =I= pj_1 and l(yj) = i + p + = pj + The algorithm 45

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puts r(xi) = min{l(yk) :Pi k n + 1 }, and by Claim 6, min{l(yk) :Pi k n + 1} = l(ypJ Let k = Pi If = pj, then we have l(yk) = + + i + p + + > pj + = l(yj), and Ix; n Iyi -::/= 0. If pj, then > pj by Claim 4, and so l(yk) = + pj + 1 + > l(yi) Again we have r(xi) > l(yj) and Ixi n Iyj -::/= 0. Now we assume pj_1 =Pi, and so l(yi) = i + p + + If Pi = n + 1, then r(xi) = l(yp;) = l(Yn+1 ) = m + 1 > l(yi) So Ixi n IYi -::/= 0. Assume Pi n. We have the following set-up: j Pi n i ClRR pj RRR I Pp, R R If pj, then > pj, and so r(xi) = l(ypJ i + p + 1 + f-(yj) If = pj, then l(ypJ = i + p + + i + p + + > i + p +! + = l(yi) So either way we have r(xi) > l(yi), which together with l(xi) < l(yi) (shown above), gives Ix; n IYi -::/= 0. This completes the proof. Figure 3.4 is the interval representation created by the algorithm in the above proof applied to the matrix A presented before Theorem 3.4 The intervals for X are black and Xi (Yi) corresponds to row i (column j). 46

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2 I 3 I 4 I 5 I 6 I Figure 3.4: Interval bigraph representation for the matrix A As another example, given the matrix 100000 111100 111010 f--7 111001 010000 1RRRRR 1111RR 111C1R 111CC1 C1CCCC we get the following interval representation and graph in Figure 3 .5. The graph is taken from a set of forbidden subgraphs determined by Muller in [37) called insects which we discuss below. An insect would have a vertex pendant to vertex 3 ofY. We will now record the results that, at the time of writing this thesis, characterize interval bigraphs. Chronologically speaking, Corollary 3.5 preceded Theorem 2.6, see [6), and was then generalized to interval k-graphs, see [7], but here we record it as a corollary. Recall that a set of distinct induced subgraphs Q = { G1 ... Gt} of a graph G = (V, E) is consecutively ordered when for any v E V, if i < j < l and v E Gin G1 then v E Gi. We say that g covers G if it forms an edge cover of G. 47

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4 2 6{}-----40----___::n Figure 3.5: An almost-insect's interval bigraph representation Corollary 3.5 A bipartite graph G is an IBG if and only if there exists a cover for G consisting of bicliques that can be cons e cutively ordered. Proof: An IBG is an interval 2-graph. Thus, by Theorem 2.6, there is a cover of complete 1-or 2-partite subgraphs that can be consecutively ordered. And conversely if there is such a cover G is an interval 2-graph also by Theorem 2.6. A circular arc graph is the intersection graph of arcs of a circle, see Figure 3.6 for an example. The clique cover number of a graph G is the minimum size of a collection of induced complete subgraphs that contain all vertices of G. For example, the complement of any connected nontrivial bipartite graph has clique cover number 2. For short, we will say that G is a k-clique graph and mean that it has clique cover number k. We have the following list of equivalences for bipartite graphs. The equivalences (a), (b), (e) and (f) wer e found by Hell and Huang, see [24). Das et al. in [10) characterized interval digraphs with (d). Theorem 3.6 The following statements are equivalent for a bipartite graph G = 48

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Figure 3.6: A circular arc graph. Figure 3.7: Forbidden configuration for Theorem 3.6 (e). (X, Y, E) with n vertices: (a) G is an interval bigraph; (b) G is a 2-clique circular arc graph in which no two arcs cover the whole circle : (c) There is a cover of G consisting of bicliques that can be consecutively ordered; (d) The reduced adjacency matrix of G has a zero-partition; (e) The vertices of G can be ordered v1 < v2 < < Vn so that there do not exist a < b < c with Va, vb in the same partite set and VaVc E E, but vbvc E (see Figure 3. 7); (f) The vertices of G can be ordered v1 < v2 < < Vn so that there do not exist a < b < c < d with the any of the four structures in Figure 3. 8. Now we prove Theorem 3.3 using the results we have developed. Assume G is a cycle-free graph. If G has NLlO as an induced subgraph, then G has an 49

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2 3 4 Figure 3.8: Forbidden configurations for Theorem 3.6 (f). ATE and hence is not an IBG, by Theorem 2.10. Now assume that G has no induced NL10. Then there is a path P from which all vertices have distance at most two. Any vertex at distance one or two from P and not on P belongs to a biclique isomorphic to K1 ,2 or K1 1 Beginning at one end of P take the edges of P one at a time as induced bicliques, and when a vertex of degree greater than or equal to three is encountered, take all of the K1 1 's and K1,2's incident with that vertex in any order and then continue along P in the same fashion. This creates a biclique cover that is consecutively ordered. Therefore, by Theorem 3.6 G is an IBG. 3.3 Forbidden Substructures and Subgraphs According to the literature, cycles of even length greater than 4, the graphs of Figure 3.10 and their generalizations, the three graphs in Figure 3.11, the insects of Figure 3.12, and the bugs of Figure 3.13 are all of the bipartite graphs that are forbidden subgraphs for IBGs that authors are willing to give explicitly. We present these graphs in turn, and explain why or give references in which it is explained why they are forbidden. We reiterate that this exposition will not describe all forbidden subgraphs and substructures for IBGs since such a list is not known. 50

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Define the neighborhood of an edge e of a graph G to be N(e) = UvEV(e) N(v). Recall that an asteroidal triple of edges (ATE) of G is a set of three edges with a path between any two that avoids the neighborhood of the third edge. In Figure 2.5 there are two bipartite graphs each with an ATE, and in Figure 3.10 the graphs Di, Wi and 1i, fori = 1, 2, 3, have ATEs induced by the thicker edges. Theorem 2.10 gives the following corollary, which was also proved in [37]. Corollary 3. 7 If G is an IBG, then there is no asteroidal triple of edges in any induced subgraph. It turns out that the principle behind ATEs can be generalized to generate many forbidden subgraphs for IBGs. Such a structure will be defined later and can be found in graphs obtained from some of the finite families of finite sets in Figure 3.9. Note that there are six infinite collections Ci, Ti, Wi, Di, Mi, Ni, and three separate families, f1 f2 f3 in Figure 3.9. Obtain a bipartite graph from a family :Fin Figure 3.9 as follows. Let :F = { Ai : 1 :::; i :::; k} consist of subsets of {1, 2, ... n }, n E z+. Represent any :F by a bipartite graph G = (X, Y, E) with X = {x1, x2, ... Xn}, Y = {YI, Y2, ... Yk}, and XiYj E E if and only if i E Aj. See Figures 3.10 and 3.11 for representations of the families as bipartite graphs. Note that the representation of any Ci is simply a cycle of even length greater than or equal to 6; C1 is a 6-cycle, C2 is an 8-cycle, and so on. The relevance of this list will become clear once the following result of Trotter and Moore is considered along with Theorem 3.6 (b.). Also, we record the list in Figure 3.9 here because it is useful for later reference; the order in which the sets are given is, in a sense, as close to being conducive to an interval bigraph representation or zero partitionable matrix as possible. 51

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---------------------Theorem 3.8 (Trotter, Moore, 1979 [36]) A graph G is a 2-clique circular arc graph if and only if G contains none of the bipartite graphs derived from the finite families of finite sets in Figure 3. 9. So, by Theorem 3.6 (b.) IBGs are the complements of restricted 2-clique circular arc graphs, and hence every graph obtained from a family in Figure 3.9 is not an IBG by Theorem 3.8. It is easy to see that any of the graphs of Figure 3.10 is minimal as a forbidden subgraph: simply remove a vertex and obtain an interval bigraph representation. We illustrate with an example, but instead of constructing an interval representation, we will take a set from Figure 3.9, remove a set and show that its incidence matrix has a zero partition. Realize .that this is equivalent to exhibiting a zero partition for the reduced adjacency matrix for the corresponding bipartite graph. Consider N3 of Figure 3.9. Re move the the set {10} giving = { {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5}, {1, 2, 3}, {1 }, {1,2,3,4,5,6,8, 10}, {1,2,3,4,6,8},{1,2,4,6},{2,4},{2,9}}; now make an in cidence matrix with row i corresponding to element i and column j correspond ing to set j with respect to the order in which they are given We get the following matrix which exhibits a zero partition after swapping rows 9 and 10: 52

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c1 = { {1, 2}, {2, 3}, {3, 1}} c2 = {{1,2},{2,3},{3,4},{4,1}} c3 = {{1,2},{2,3},{3,4},{4,5},{5,1}} T1 ={{1,2l,{2,3},{3,4},{2,3,5},{5}} T2 = {{1,2 ,{2,3 ,{3,4},{4,5},{2,3,4,6},{6}} T3 = {{1,2 7},{7}} w1 = { {1, 2}, {2, 3}, {1, 2, 4}, {2, 3, 4}, { 4}} w2 = {{1,2},{2,3},{3,4},{1,2,3,5},{2,3,4,5},{5}} W3={{1,2},{2,3},{3,4} {4,5},{1,2,3,4,6},{2,3,4,5 6},{6}} 1)1 ={{1,2,5},{2,3,5},{3},{4,5},{2,3,4,5}} 1)2 = {{1,2,6},{2,3,6},{3,4,6},{4},{5,6},{2,3,4,5,6}} 1)3 ={{1,2,7},{2,3,7},{3,4,7},{4,5,7},{5},{6,7},{2,3,4,5,6,7}} M 1 = {1, 2, 3, 4, 5}, {1, 2, 3}, {1 }, {1, 2, 4, 6}, {2, 4}, {2, 5}} M2 = {{1,2,3,4,5,6, 7},{1,2,3,4,5},{1,2,3},{1},{1,2,3,4,6,8}, {1,2,4,6},{2,4},{2, 7}} M3 = {{1,2,3,4,5,6, 7,8,9},{1,2 3,4,5,6, 7},{1,2,3,4,5},{1,2,3},{1}, {1,2,3,4,6,8,10},{1,2,3,4,6,8},{1,2,4,6},{2,4},{2,9}} N 1 = { {1, 2, 3}, {1 }, {1, 2, 4, 6}, {2, 4}, {2, 5}, {6}} N2 == { {1, 2, 3, 4, 5}, {1, 2, 3}, {1 }, {1, 2, 3, 4, 6, 8}, {1, 2, 4, 6}, {2, 4}, {2, 7}, {8}} N3 = { {1,2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5}, {1, 2, 3}, {1 }, {1, 2, 3, 4, 5, 6, 8, 10} {1,2,3,4,6,8},{1,2,4,6},{2,4},{2,9},{10}} r1 = { {1, 3, 5}, {1, 2}, {3, 4}, {5, 6}} r2= {{1},{1,2,3,4},{2,4,5},{2,3,6}} r 3 ={{1,2},{3,4},{5},{1,2,3},{1,3,5}} Figure 3.9: The finite families of finite sets that represent the forbidden com plements of 2-clique circular arc graphs. 53

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N3 NI 3 ei 123 2 25 5 4 24 e3 M3 e1 eo e3 e4 es I ei e2 e3 D1$n Figure 3.10: Some of the corresponding graphs of the families in Figure 3.9. The thicker edges are ATEs or edge asteroids. The black vertices represent elements and the white vertices represent subsets. 54

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4 6 2 t-1 123 )-56 5 4/ 3 t---' )-. 245 236 34 -2 34 3 135 1 12 2 5 ( 6 4 rl r2 I3 Figure 3.11: The graphs corresponding to r 1 r2, r3 = 11111110. 0 111011111 111011000 110011110 110010000 100011100 r-+ 100000000 000011000 000000001 000010000 1 1 1 1 1 1 1 RR 1 1 1 c 1 1 1 1 1 111C11RRR 11CC1111R 11CC1RRRR 1CCC111RR 1CCCRRRRR CCCC11RRR CCCC1RRRR CCCCCCCC1 therefore is an interval bigraph. 12 135 5 5 The principle behind an ATE generalizes and the generalization is what prohibits the bipartite graphs derived from Mi and Ni from being IBGs. This structure was identified and defined by Feder, Hell, and Huang in [12] as a generalization, or perhaps a modification, of the principle behind the asteroids defined by Gallai in [14] that helped him characterize transitively orientable graphs. We will not define Gallai's asteroids, but we note that we will deal with 3-asteroids, a.k.a asteroidal triples, when we turn our attention to unit probe interval graphs and unit interval k-graphs. Let G = (X, Y, E) be a bipartite graph. An edge asteroid in G is a set of 2k + 1 edges e0 e1 ... e2k and 2k + 1 55

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paths P0 1 P1,2, ... P2k,o where each Pi,i+l joins ei to ei+l and contains ei and ei+l, so that for each i = 0, 1, ... 2k the path Pi+k,i+k+l does not intersect N(ei), and subscripts are modulo 2k + 1. If an edge asteroid consists of 2k + 1 edges, we call it a 2k +l-edge asteroid. In [12] Feder, Hell, and Huang give a new characterization of 2-clique circular arc graphs as precisely those 2-clique graphs that have no induced cycle of length 6 or greater and no edge-asteroid in their complements. For examples of edge asteroids, we refer to the graphs M1 M2 and N2 of Figure 3.10. M1 has a 5-edge asteroid because there is a path between e2 and e3 that avoids the neighborhood of e0 ; there is a path between e3 and e4 that avoids the neighborhood of e1 ; we can find a P4 0 that avoids N(e2); there exists P0 1 avoiding N(e3); and finally, there is a path from e1 to e2 that avoids N(e4 ). Now, in M2 and in N2 there is a 7 edge asteroid. So, fori= 0, 1, ... 6, using the labeling in the figure and noting that, in the parlance of the definition above, k = 3, there is a path from ei + 3 to ei+3+1 that avoids N(ei) There is a subtle difference between an ATE and a 3-edge asteroid, and it can be seen using a chordless 6-cycle. Take every other edge of a 6-cycle, call them e0 e1 e2 These edges constitute an ATE. Now, fori= 0, 1, 2 and taking subscripts modulo 3, the definition of a 3-edge asteroid requires that the path Pi+l,i+ 2 from ei+l to ei+ 2 include ei+l and ei+2, and so N(ei) n Pi+l,i+2 =I 0. Hence, a 3-edge asteroid is an ATE, but not always conversely. Every family of Figure 3.9 generates a bipartite graph with either an ATE or an edge asteroid, see [12]. The families Ci, 1i Wi and Di fori;::: 1 have ATEs while Mi and Ni for i ;::: 1 have edge asteroids: M1 and N1 each have a 5-edge 56

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asteroid; M2 and N2 each have a 7-edge asteroid, and so on. Essentially Mi or Ni is obtained from Mi-l or Ni-l, respectively of course, by adding 4 vertices that induce 2 edges that become members of the (2(i + 1) + 1)-edge asteroid in Mi or Ni Theorem 3.8 and Theorem 3,6 imply that the determination of the complete list of forbidden subgraphs for IBGs rests on the determination of characterizing those bipartite graphs whose complements are 2-clique circular arc graphs, but require 2 arcs to cover the host circle in any representation. To this end the classes of bipartite graphs called insects are defined in [37]; and bugs are defined in [24). An insect is any graph isomorphic to one of the three in Figure 3.12. The bugs are given in Figure 3.13. In [37], Muller conjectured that a bipartite graph is an IBG if and only if it has no ATE, and no induced insect. The connection between IBGs and circular arc graphs was not known at the time of Muller's conje cture. Indeed, the graph N1 in Figure 3 .10 has no chordless cycle of length greater than 4, no ATE, and no induced insect, but is not an IBG by Theorems 3.8 and 3.6 (b.) and so the conjecture is disproved. However, it is easy to verify that the complement of an insect is a 2-clique circular arc graph, but any representation requires two arcs to cover the host circle, see [24). We present a definition, or a structure if you will, that unifies the insects and bugs. It was defined by Hell and Huang in [24). Let G = (X, Y, E) be a bipartite graph. For vertices x1 x2 of G, we say that N(x1 ) and N(x2 ) are comparable if either N(x1 ) N(x2 ) or N(x2 ) N(x1). An exobiclique of G is a biclique induced on M X and N Y such that X\ M and Y \ N each contain three vertices with incomparable neighborhoods contained in the biclique. 57

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Figure 3.12: The three insects. Figure 3.13: The bugs. 58

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X y Z i Y-N :x-M v w: u Figure 3.14: The exobiclique of a bug. Black vertices represent X. Figure 3.14 illustrates this definition: N(x), N(y), N(z) and N(u), N(v), N(w) are incomparable. The vertices of degree one and two in each graph of Figure 3.12 constitute exobicliques, as do the vertices of degree two and three in each graph of Figure 3.13. Theorem 3.9 (Hell, Huang, 2003 [24]) If a bipartite graph G contains an exo biclique, then in any representation of G by circular arcs, there are two arcs that together cover the whole circle, and hence G is not an IBG. 3.3.1 Concluding Remarks The most prescient finding in all of the recent work on interval bigraphs is their connection with circular arc graphs. Indeed, this mode of exploration has not been exhausted, at least not by the author. In particular, the results for circular arc graphs suggest the idea of characterizing IBGs via their complements. Sometimes a characterization is easier to produce or discover by way of considering the complementary situation. The result of Trotter and Moore 59

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._,, above, and the work of Gallai on transitively orientable graphs, shows that it is sometimes advantageous to proceed in this fashion. In regards to circular arc graphs, we will see that they are in fact intimately tied to IBGs and all of the subclasses of IBGs we consider in this thesis. Namely, we will see that proper circular arc graphs correspond precisely to the comple ments of unit interval bigraphs, and that the interval-point bigraphs are the complements of another class of circular arc graphs. In summary: The IBG forbidden subgraph characterization remains an open problem. 60

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4. Interval Point Bigraphs Suppose G is an interval bigraph with the property that, in some represen tation, one of the families of intervals is restricted to consist of points only; call G an interval point bigraph. We investigate interval point bigraphs, characterize them via the structure of their reduced adjacency matrices, show that they are precisely the complements of a certain class of 2-clique circular arc graphs in the fashion of the result for interval bigraphs by Hell and Huang, and identify them with a class of bipartite probe interval graphs. We also develop a conjecture for a forbidden subgraph characterization that is supported by the result of Tucker in [45]. 4.1 Background Let G = (X, Y, E) be an interval bigraph and (Ix,Iy) its interval repre sentation. We investigate what happens when we restrict one of the families of intervals Ix,Iy to be points only. Formally, we define an interval point bigraph to be the bipartite intersection graph G = (X, Y, E) with one partite set in one-to-one correspondence with a collection of points, P = {Pu}, and the other partite set in one-to-one correspondence with a family of intervals I= {Iv} with uv E B whenever Pu E Iv. Clearly, any interval point bigraph is an interval hi graph. We note at the outset that if G = (X, Y, E) is an interval point bigraph, then the choice of which partite set is to correspond to a collection of points de pends on the structure of the graph. We make some notation to emphasize this issue. First of all, for the purposes of this chapter, we think of a bipartite graph 61

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G = (X, Y, E) as an ordered triple in which X is in the first slot, Yin the second, and E in the third. The partite set that corresponds to the collection of points is called the point partition. So, if G = (X, Y, E) is an interval point bigraph and X is the point partition, this is denoted G = (P, I, E), and linguistically by "G is a PIBG" (PIBG is a mnemonic for "point interval bigraph"); similarly, with Y the point partition, we have G = (I, P, E), and linguistically "G is an IPBG". We use "interval point bigraph" if we wish to make no specification as to which of X, Y is the point partition, that is, if we wish to speak about the class of all bipartite graphs that are either IPBGs or PIBGs. The collection of points and intervals representing an IPBG or a PIBG is called the interval point representation. Figure 4.1 shows three bipartite graphs each with an interval point representation to its right. Recall that a directed graph is an interval digraph if two intervals S(x) and T ( x) of lR can be assigned to each vertex x such that u -+ v if and only if S ( u) n T(v) 1-0. An interval point digraph is an interval digraph with T(x) a singleton point for each vertex x. In [10] interval point digraphs were characterized via their adjacency matrices: D is an interval point digraph if and only if A(D) has the consecutive 1's property for rows, where A(D) = (ai,j), with ai,j = 1 if vi -+ vi and 0 otherwise. Again, we will see that form the perspective of bipartite graphs we are able to see more structure, than from the directed graph perspective. In this chapter we characterize interval point bigraphs in terms of the structure of their reduced adjacency matrices and show that they are equivalent to a certain class of bipartite probe interval graphs. Also, along the lines of the 62

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9 10 2 7 4 10 7 8 9= 8= 1= 1 2 3 4 5 3 5= ':i ','! 8 2 4 6 9 Hll 1= 3 0 0 5 1 2 3 4 5 6 7 7= 10= 11= ::r 1 3 5 7 10 9I 2= 8= 4= 12= ..--.-a 0 0 6= 1 2 3 4 5 6 7 8 9= Figure 4.1: Three IPBGs, dark vertices belong to the point partition result of Hell and Huang identifying interval bigraphs with a class of restricted circular arc graphs, we determine which class of circular arc graphs correspond to interval point bigraphs. Recall that a component problem to determining whether a given graph is a probe interval graph is in determining how to partition the vertices into probes and nonprobes. We will show that interval point bigraphs correspond precisely to the class of bipartite probe interval graphs in which the probe/nonprobe partition is also a bipartition. A list of forbidden subgraphs is given and we conjecture that this list provides a forbidden induced subgraph list for interval point bigraphs. Following the analogous results for interval graphs, probe interval graphs, and interval k-graphs, we give a consecutive subgraph characterization for interval point bigraphs. Finally, following 63

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the analogous results for unit interval bigraphs and interval bigraphs, we show that complements of interval point bigraphs correspond precisely to a restricted class of 2-clique circular arc graphs. 4.2 A Matrix Characterization Recall that if G = (X, Y, E), lXI = m IYI = n, then we represent Gas a (0, 1)m x n matrix A( G) = (aij) with aij = 1 if and only if XiYi E E, and throughout this thesis, rows of A(G) are indexed by X (the partite set in the first slot). A (0, 1)-matrix M has the consecutive 1 's property for columns if the rows can be permuted so that the 1 's in every column appear consecutively; the consecutive 1 's property for rows is defined similarly. Theorem 4.1 A bipartite graph G = (X, Y, E) is an interval point bigraph if and only if its reduced adjacency matrix has the consecutive 1 's property for rows or for columns. Proof: Let M = (mij) be an m x n(0, 1)-matrix with the consecutive 1's property for rows or for columns. For the sake of argument, we will assume that M is found to have the consecutive 1 's property for rows, since the other case for columns is symmetric. With M exhibiting consecutive 1 's in the rows, we will build an interval point representation for the bipartite graph G(M) that M represents. Call then vertices corresponding to columns {y1 ... Yn} = Y and them vertices corresponding to rows {x1 ... Xm} =X. Create a set of points P = {p(yl), ... P(Yn)} corresponding to Y with p(yl) < p(y2) < < P(Yn). Create a collection of intervals T = {I(x1), ... I(xm)} corresponding to X with I(xi) [l(xi), r(xi)], l(xi) = min{j : mij = 1)}, and r(xi) = max{j : mij = 1 }. This yields G(M) = (X, Y, E) with XiYj E E if and only if p(yj) E I(xi) since 64

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M has the consecutive 1's property for rows. G(M) is an IPBG, as desired. Now, let G = (X, Y, E) = (P,I, E) be a PIBG. We will show that A( G) = ( a i j) has the consecutive 1 's property for columns; that A( G) has the consecutive 1's property for rows if G is an IPBG follows from a similar argu ment. We may assume that the points of P are distinct in the representation so let P = {p(x1), ... ,p(xm)} where the indexing is done so that p(xi) < p(xj) when i < j. In A(G) make row i correspond to xi Because each Yi E Y is represented by an interval, I(yj), the neighborhood of any Yi must be a con secutive set with respect to the order given by the point representation. Make column j of A( G) correspond to Yi and observe entry ( i, j) = 1 if p(xi) E I(yj) A(G) exhibits consecutive 1's in its columns because if p(xa),p(xc) E I(yj), then p(xb) E I(yj), for a < b < c; that is, if aaj = acj = 1, then abj = 1 for any a< b
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be ordered Xa < Xb for indexes a < b such that for each vertex y E Y, N(y) is a consecutive set with respect to <; i.e., if XaY E E and XcY E E, then XbY E E for any a < b < c. A Y -consecutive bipartite graph is defined similarly. Clearly, if G is X-consecutive, then A(G) has the consecutive l's property for columns; similarly, if G is Y-consecutive, then A(G) has the consecutive l's property for rows. In [45] Tucker determined what is necessary and sufficient for a (0, I)-matrix to have the consecutive l's property for columns. It could be said that he studied this problem because of the frequency a consecutive ordering anomaly comes to bear in many problems, especially with intersection graphs having interval models. Indeed, we have many instances in this thesis in which a consecutive ordering of some collection of subgraphs is either necessary or sufficient or both. Another structural property that is apparently important in the study of intersection graphs of interval sort is whether the graphs in question contain asteroidal triples. Recall that an asteroidal triple (AT) of a graph is a set of three vertices with a path between any two that avoids the neighborhood of the third. The next theorem of Tucker's shows why we must choose the darkened vertices of Hll in Figure 4.1 as the point partition, and relates ATs to consecutive order properties. Theorem 4.2 (Tucker, 1972, [45]) A bipartite graph G = (X, Y, E) is Xconsecutive if and only if it has no asteroidal triple of G wholly contained in X. In Hll, the sets {1, 10, 5} and {3, 11, 7} are ATs and both sets are contained in one of the partite sets, and that partite set cannot correspond to points in an interval point representation. Thus, the problem of determining whether a given 66

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bipartite graph is an interval point bigraph lies, in part, in determining which set to use as the point partition. Thcker's result, Theorem 4.2, gives the following theorem which is prescient to our classification of interval point bigraphs. Corollary 4.3 A bipartite graph G = (X, Y, E) is an interval point bigraph if and only if there is no AT of G contained in X or there is no AT of G contained .in Y. Proof: If G = (X, Y, E) has no AT in X, then it is X-consecutive by Theorem 4.2, and so A( G) has consecutive 1 's in columns and so G is a PIBG by Theorem 4.L Similarly, if G has no AT in Y, then G is Y-consecutive and A(G) has consecutive 1's in rows; so, G is an IPBG. If G is a PIBG, A( G) has consecutive 1's in rows by Theorem 4.1 and G is therefore Y-consecutive. Similarly, if G is an IPBG, then A( G) has consecutive 1 's in the columns; therefore, G is X -consecutive. So, if we were to determine all the minimal bipartite graphs that have an as teroidal triple in each of the partite sets, then we would have a complete list of forbidden induced subgraphs for interval point bigraphs. The way Tucker characterized the consecutive 1's property for columns in (0, 1)-matrices was by looking at the bipartite graphs that represent the matrices. In our terminology, he characterized the X -consecutive bigraphs by forbidden subgraphs via Theorem 4.2. Here is his result. Theorem 4.4 (Tucker, [45]) A bipartite graph G = (X, Y, E) is X -consecutive if and only if it has no subgraph isomorphic to any of the graphs in Figure 4.2, 67

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where darkened vertices represent X. Note that there are three infinite families in Figure 4.2: C(n),!In, and IIIn, where C(n) is simply all cycles with even length greater than 4; and we have given examples of II1,II2,Ih and IIh,III2,IIh_and III4 Also note that II1 and I I In for n 3 are IPBGs, so this list does not serve as a complete forbidden subgniph list for interval point bigraphs. To see that II1 III1 III2 and III3 are IPBGs, refer to Figure 4.3 and see that each has a reduced adjacency matrix with consecutive 1 's in the rows. Referring to Figure 4.2, I I !4 is not an interval ...... point bigraph because the sets { x, y, z} and { c, d, e} are ATs, and observe in I 12 and I !3 the sets {x, y, z } and {a, 1, 2} are ATs. Next we give our conjecture for a complete list of forbidden induced sub graphs for interval point bigraphs. The necessity of the conjecture is easy to verify: simply locate an asteroidal triple in each partite set. Conjecture 4.5 A bipartite graph G is an interval point bigraph if and only if G has no induced subgraph isomorphic to any in Figure 4.4, or C(n),IIIn (for n 2: 4), I In {for n 2: 2}, II;, or NL10 of Figure 4.2. 4.4 A Consecutive Order Characterization Recall that a collection Q = { G1 ... Gm} of subgraphs of a graph G is consecutively ordered if v E Gin Gk ===} v E Gi for Gi < Gi < Gk If for each e E E(G), e E E(Gi), for some Gi E Q, then Q is a cover for E(G), or Q covers E(G); if there is exactly one Gi containing e for each e E E(G), then Q is a partition for E(G), or Q partitions E(G). We have characterizations for interval k-graphs via a cover of consecutively orderable complete multipartite 68

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X={} 2 III II a 2 a I 2 3 y z b c d e Figure 4.2: Forbidden X -consecutive bigraphs. subgraphs, and in particular, a characterization for interval bigraphs via a cover of consecutively orderable bicliques. Also, we have a characterization for probe interval graphs via a consecutively orderable cover of maximal quasi cliques, 69

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d b a c X 0 1 1 0 y 1 1 0 0 z 0 0 1 1 w 1 1 1 1 a b c X 1 0 0 y 0 1 0 z 0 0 1 w 1 1 1 y b w c z a b c d X 0 1 0 0 y 1 0 0 0 z 0 0 0 1 1 I 1 I 0 2 0 I 1 1 a c d a b d c e X X 0 0 1 0 0 d y 1 0 0 0 0 z 0 0 0 0 I 1 I 1 I 0 0 y 2 0 I I I 0 3 0 0 I I 1 z a b c e Figure 4.3: IPBGs from Tucker's list. given by Theorem 2.4. Recall that a maximal quasi clique Q = (Po, No) in a probe interval graph H is a subset P0 c V(H) of probes that form an induced complete subgraph of H, together with a (possibly empty) set N0 c V(H) \Po of nonprobes, each adjacent to every probe in P0 Our next result gives a characterization for interval point bigraphs in terms of a consecutive ordering of stars that form a partition of E( G), and hence indicates the distinction between interval point bigraphs and interval bigraphs. A star is a K1,n, for n 0, and the center of a star is the partite set of size 1 (for a K1 1 either vertex may be thought of as the center). 70

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... k" p I 2n 11 0 2nA Figure 4.4: Forbidden interval point bigraphs; dashed edges may or may not be present Theorem 4.6 A bipartite graph G = (X, Y, E) is an interval point bigraph if and only if it has a consecutively orderable edge partition of stars with all centers in the same partite set. 71

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Proof: Let G = (X, Y, E) be an IPBG; the case for G a PIBG is similar. By Theorem 4.1, A(G) = (aij) exhibits the consecutive 1's property for rows. Let ai be the first column in which a 1 appears in row i. Permute rows of A( G) so that { ai} forms a nondecreasing sequence. Now, with Yi E Y and xi E X corresponding to column j and row i, respectively, take each Yi as the center of star sj and put sj = {yj} u {Xi : XiYj E E}. That is, take the star given by each column of the matrix A( G) for which { ai} forms a nondecreasing sequence; the order of the columns gives the order of the stars. To see that this ordering is consecutive, it suffices to check that Xi E San Be ====} Xi E sb for any a < b < c. If Xi E Sa n Be, but Xi sb, then aia = 1 = aae, but aib = 0, contradicting the fact that A( G) has consecutive 1 's in the rows. Conversely, letS= { 31 ... Sr} be a partition of G = (X, Y, E) consisting of stars with indexing corresponding to the consecutive order and so that each star has its center in X. We will show that G is a PIBG, but note that if the centers all belonged to Y, then we would obtain an IPBG from the analogue of the following construction. Make a collection of points P = p(xr) < p(x2 ) < < p(xr), where xi is the center of star Si. Now, for each y E Y, make I(y) = [l(y), r(y)], where l(y) =min{ i : y E Si} and r(y) =max{ i: y E Si} We have p(xi) E I(y) if and only if Xi and y are both contained in some star together which happens only if XiY E E. Thus, the collection of intervals and points is an interval point representation for G. Theorem 4.6 indicates that what distinguishes an interval point bigraph, and hence, as we will see, certain probe interval graphs and the complements of 72

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certain 2-clique circular arc graphs, from an interval bigraph is precisely the ability to partition the edges of the graph into stars as opposed to needing to use 4-cycles and larger bicliques to cover its edges. An open and perhaps interesting problem would be to characterize those bipartite graphs in which one is forced to repeat edges in any consecutive cover, and hence cannot consecutively partition the graph. 4.5 Probe Interval Graphs and Interval Point Bigraphs The problem of choosing which partite set is to correspond to points, if in fact the graph being considered is an interval point bigraph, is similar to the problem of choosing how to partition vertices into probes and nonprobes when determining whether a given graph is a probe interval graph. If the partition into probes and nonprobes is given, then the recognition problem is solved easily, that is, there are recognition algorithms that have polynomial running times, see [28]. But when the partition is not given, the complexity of the recognition problem is not known. As a simple example illustrating the issue of partition choice, and for an example that speaks to our next results, consider H12 of Figure 4.5. In a probe interval representation, vertices c and f must be nonprobes, but they belong to different partite sets in the bipartition of Hl2. Also, H12 is not an interval point bigraph because {a, k, e} and { d, l, h} are both ATs, each belonging to a different partite set. The next result, especially in light of Theorem 2.6 and Theorem 4.6, gives more precision to where bipartite probe interval graphs lie with respect to interval point bigraphs and interval bigraphs. 73

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k c d e b a H12 f g h k= iac e-I= j-hg-b-d-f '==== Figure 4.5: Example of a bipartite PIG with P, N partition not a bipartition. Theorem 4. 7 If G is a bipartite probe interval graph, then G has a consecutively orderable partition of stars. Proof: Let G = (P, N, E) be a bipartite probe interval graph. Theorem 2.7 shows that G is an interval bigraph. By Theorem 2.4, G has an edge cover maximal quasi cliques that are consecutively orderable. By definition, any max imal quasi clique of G is either a K2 c P or a star with center in P and all other vertices in N. Hence, the complete set of maximal quasi cliques Q consists of stars, say Q = {81, ... Sm}, where the indexing gives the consecutive order. By Lemma 2.5, Q can be made so that each maximal clique of G is in exactly one Qi E Q. Hence, Sin Si+l cannot contain an edge, since an edge is a maximal clique in a bipartite graph. Therefore Q forms a consecutive edge partitoin of stars .. The converse of Theorem 4.7 is not true: consider HlO in Figure 4.6. HlO is not a PIG because, after creating an interval representation for a longest path, we see that if either of the vertices of degree three is a probe, there is no way to place the vertex at distance two from the path. But HlO has a consecutive 74

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partition consisting of stars. HlO -: : : : 4 L ... .t : i : : 21 I .......... ....... li : l ........ ........ :_--r ------------:. : r : f -=t--+;:............ ......... 1 :-----;-----__ : 3 Figure 4.6: HlO has a consecutive partition into stars, but is not a probe interval graph. Corollary 4.8 An interval point bigraph is a bipartite probe interval graph, but the converse is not necessarily true. Proof: If G is an interval point bigraph, then by Theorem 4.6 there is a partition of stars with all centers in one partite set. By defining P and N to be the centers and non-centers of the stars, respectively, we obtain a complete set of maximal quasi cliques of G that are consecutively ordered H12 of Figure 4.5 is a bipartite probe interval graph, but not an interval point bigraph. To summarize these containment relationships just mentioned, we have IPBGs u PIBGs s;; biPIGs s;; IBGs with H12 and HlO as separating examples. Next, we characterize those bipartite probe interval graphs in which the partition of vertices into probes and nonprobes corresponds to a bipartition. Also, given the appearance of more and more papers about probe interval graphs 75

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in the literature, it gives more motivation for determining the truth of Conjecture 4.5. Theorem 4.9 G = (X, Y, E) is a bipartite probe interval graph in which the probe and nonprobe assignment can correspond to the bipartition if and only if G is an interval point bigraph. Proof: Let G = (X, Y, E) be a PIBG. If we make all vertices in X probes, when X is the point partition and all other vertices nonprobes, we get G = (P, N, E) is a probe interval graph. Similarly, if G is an IPBG, then we put N=X,P=Y. Let G = (P, N, E) be a bipartite probe interval graph with P, N each corre sponding to a partite set. Let = P, {IPJ be the family of intervals corre sponding toP, and {InJ be the family of intervals corresponding toN= {nj} Since P is independent, for any i =/= j combination, !Pin Ipj = 0. Label so that l(pi) < l(pj) if and only if i < j. Now, for each nj, extend Inj so that l(nj) = l(pi) for the smallest i such that Pi E N(ni) Now shrink each !Pi to its left-endpoint and get G is an interval point bigraph in which P becomes the point-partition. 4.6 Interval-Point Bigraphs and Circular Arc Graphs Recall that a circular arc graph is the intersection graph of arcs of a circle. The circle from which the arcs are obtained is called the host circle. A graph is a 2-clique graph if its vertices can be partitioned into two cliques and all edges not contained in these cliques lie between them; i.e., the complement of a bipartite 76

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graph is a 2-clique graph. Interval bigraphs correspond to the complements of 2-clique circular arc graphs with the property that, in some representation, no two arcs cover the host circle. We explore which circular arc graphs correspond to the complements of interval point bigraphs Clearly, the circular arc graphs we seek are 2-clique graphs. Before we prove the main result of this section, we will use the properties of the reduced adjacency matrices of interval point bigraphs and the vertex ordering properties given by this characterization to characterize the complements of interval point bigraphs Another way to state Theorem 4.1 is as follows A bipartite graph G = (X, Y, E) is an interval point bigraph if X or Y can be ordered with < such that (if X is ordered) ux, uz E E ==} uy E E for u E Y, x, y, z EX and x < y < z (switch roles of X andY ifY is ordered). Given a graph G and H V (G), N H ( v) denotes the neighborhood of v restricted to the set H. For a graph G = (V, E), and X= {xll x2 .. Xm} V, X is circularly indexed if for each vinV \X, and i < j, Nx(v) is either xi, Xi+l, ... Xj or Xj, Xj+l, ... Xm, x1 ... Xi A matrix M has the circular 1 's property for columns if the rows can be permuted so that the l's in each column are circular, that is, they appear in a circularly consecutive fashion; as if the matrix were wrapped around a cylinder. The circular 1 's property for rows is defined similarly. Let us say that the 2-clique graphs in which one of the cliques can be circularly indexed have the circular indexing property, or the CIP. Our last and main result of this section, we use Theorem 4.1 and the interval point represntation to show that 2-clique graphs with the CIP are circular arc graphs, and that the complements of interval point bigraphs are 2-clique circular arc graphs with the CIP. 77

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Theorem 4.10 A bipartite graph G = (X, Y, E) is an interval point bigraph if and only if G is a 2-clique circular arc graph with the CIP. Proof: Suppose G = (X, Y, E) is an IPBG with IYI = n; we will prove that G is a 2-clique circular arc graph such that Y can be circularly indexed in G. If G were a PIBG, then G would turn out to be a 2-clique circular arc graph in which X can be circularly indexed. Since G is an IPBG, it is Y-consecutive and so we may assume that an arbitrarily chosen x EX is adjacent to Yi, ... Yk EYinG, for 1 ::::; i ::::; k::::; n, and hence, xis adjacent to Yk+l, ... Yn, Y1, ... Yi-1 in G. We construct a circular arc representation for G. Let C be a circle with two specified, diametrically opposed points p and q, with A (respectively B) the segment of C extending clockwise from p to q (respectively q to p). Let I be the interval point represetnation for G. G's structure dictates that Py1 < Py2 < < Pyn, we may use I(x) = (pi,Pk], and we assume that the points are spaced equidistantly by some constant, say c, and that the total width of I is Pyn -Py1 ; that is, the leftmost interval has left endpoint equal to Py1 and the rightmost interval has right endpoint equal to PYn. Place a copy of I in A with Py1 = p and place a copy of I in B with Py1 = q. We assume A and B are large enough so that Pyn < q in A, and Pyn < p in B. Let R( v) denote the arc corresponding to vertex v and let cc( v) denote the counter-clockwise endpoint of R( v) and cl ( v) the clockwise endpoint of R(v). Construct open R(v) = (cc(v), cl(v)) for each v E V(G) as follows. Put (cc(yi)) = Pyi E A and cl(yi) = Pyi E B. Put cc(x) = pYk E B and cl(x) = Py; E A. In this representation R(x) n R(yi) f. 0 whenever j E { k + 1, ... n, 1, ... i -1}, so Y is circularly indexed. Since x was arbitrary, this construction applied to each x E X gives a circular arc 78

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representation of G, a 2-clique graph with the CIP. Let G be a 2-clique graph with X and Y the sets inducing the cliques, and M(G) = (mij) the adjacency matrix for G; that is mij = 1 if vivj E E(G) and 0 otherwise If X can be circularly indexed, then we can permute corresponding rows and columns so that M(G) = [I X A l A IIYixiYI where I is the square matrix with O's on the diagonal and 1 's everywhere else, A= M(G)[X; Y], that is, A is the matrix induced on the rows corresponding to vertices in X versus the columns corresponding to the vertices in Y Since X is circularly indexed, A has the circular 1's property for columns, and AT has the circular 1 's property for rows. Thus, M(G) = [ where A is the reduced adjacency matrix for the bipartite graph G and will clearly have the consecutive 1's property for columns. Hence, G is a PIBG, by Theorem 4.1. Taking complements, and disregarding which clique can be circu larly indexed, we see that if G is a 2-clique graph with the CIP, then G is an interval point bigraph. In Figure 4.7 we have illustrated the idea behind the proof of Theorem 4.10 by constructing a circular arc representation for G using the construction in the proof. The figure drawing environment at the disposal of the author is not conducive to drawing arcs and circles, so we have used a square representation nothing is lost of course, since topologically circles and squares are the same. 79

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YI Y2 Y3 G x3 0 0 0 XI -Y2 X2 X3 -x4 -XI YI x2 Y3 x4 interval point representation for G circular arc representation for G Figure 4. 7: A circular arc representation of the complement of an interval point bigraph using the method in the proof of Theorem 4.10. The dashed lines are meant to indicate the arcs. 80

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We conlude with the following list of equivalences for bipartite graphs. Theorem 4.11 Let G = (X, Y, E) be a bipartite graph. The following are equiv alent: ( 1.) G is a bipartite probe interval graph in which the probejnonprobe partition can correspond to the bipartition; {2.) G is a 2-clique circular arc graph in which one of the cliques can be circularly indexed; (3.) G is an interval point bigraph; (4.) G has a consecutively orderable edge partition of stars in which all centers belong to the same partite set. 81

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5. Unit Interval k-graphs Results in this chapter have arisen from restricting attention to the case where all intervals have the same length. We have found this restriction, as it forces quite a bit of structure, to be fruitful in terms of discovering structural properties. Unit interval bigraphs for example are shown to be equivalent to over a dozen other classes of structured graphs and structural properties. Also, like the unit interval graphs, see [39], and unlike tolerance graphs, see [2], unit inter val bigraphs are equivalent to proper interval bigraphs. The characterizations include one of forbidden substructures, and this characterization gives a struc ture theorem for a class of structured (0,1)-matrices similar to the way Theorem 4.4 gives a structure theorem for the consecutive 1 's property for columns. 5.1 Background We report on findings obtained while determining a forbidden subgraph characterization for unit interval bigraphs, and unit interval k graphs. The results for unit interval bigraphs involve several other classes of graphs some of which we now define. Some definitions however are not given in this background section, but are defined closer to where they are needed. Recall an interval k-graph (IkG) is the k-partite intersection graph of k distinct families of intervals F = {I1,I2 ... ,Ik}, with vertices adjacent if and only if their corresponding intervals intersect, and each interval belongs to a distinct family. An IkG is a unit interval k-graph (UikG) if in some repre sentation all intervals can be made to have identical length. For G an IkG, or 82

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UikG, recall from Chapter 2 that the interval classes correspond to independent sets of G and are essentially obtained from a proper coloring of G. We also re marked in Chapter 2 that not every proper coloring is conducive to an interval k-representation, for example giving every vertex a distinct color works only if G is an interval graph. But we also showed that IkGs are perfect and in do ing this showed that using w(G) interval classes will always suffice. But within the class of UikGs, we have found anomalous occurrences in which only a very specific w( G)-coloring corresponds to an interval class assignment conducive to a unit interval k-representation. Figure 5.1 serves as an example of this. The graph of Figure 5.1 is a UikG, as evinced by the top representation, but with the bottom interval class assignment, there is no way to place 1(6) so that it does not intersect 1(2) or 1(4), which violates the fact that vertices 2 and 6, and 2 and 4 are not adjacent. Thus, a characterization of UikGs will require the understanding of this and perhaps the invention of a very specific coloring scheme. The example suggests that the scheme must take into account not only the neighbors of a vertex, but also the neighbors at distance 2. If no interval properly contains another, in some interval k-representation, then the graph corresponding to the representation is a proper interval k-graph. The vast majority of this chapter is devoted to the bipartite unit interval kgraphs, that is, the unit interval bigraphs (henceforth UIBGs), and the proper interval bigraphs (henceforth PIBGs). We will now define a few of the notions and classes of graphs we will explore. The following class of bipartite graphs turns out to be equivalent to the class of unit interval bigraphs, and has been useful in giving the authors helpful 83

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1 2 3 4 5 1 3: 5:--. ---------t----t-----4 --:---------: : 7 : : . 6 7 . . 2 . 1 2 3 4 5 6 7 !3---.. 1---5--, ... --:----------:-, :6; ; : '--jo----------;..1 : : Figure 5.1: A UikG in which the ability to represent it rests on using a particular interval class assignment. The horizontal dashed lines distinguish the interval classes (as do their shadings), the vertical dotted lines serve to help determine intersection. perspectives. A valuation bigraph, G = (X, Y, E), is a bipartite graph, with functions f g: Y--+ that if x EX, y E Y, then xy E E if and only if If ( x) g (y) I 1. The functions f, g are called the valuation functions. In other words, a valuation bigraph is the bipartite analogue of a valuation kgraph discussed in Chater 1. We discuss valuation k-graphs again briefly as a prelude to Theorem 5.1, and reiterate that the bipartite graph perspective, and in particular the perspective given by valuation bigraphs, has proved to be fruitful for us. An asteroidal triple (AT) is a set of three vertices with a path between any two that does not intersect the neighborhood of the third. A graph that has no asteroidal triple in any induced subgraph could be thought of as a graph that grows in only two directions. The interval graphs have been shown to be equivalent to the class of graphs that do not contain an asteroidal triple on 84

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any induced subgraph, or an induced cycle of length at least four, see [4]. AT free graphs have been studied extensively perhaps because of the appearance of ATs as forbidden substructures in many classes of graphs; they have been characterized in [9]. It will turn out that all of the bipartite graph classes we investigate in this chapter are precisely those bipartite graphs that are AT-free. The class of graphs we define next is not always thought of as a class of intersection graphs, but we present the definition which renders them as such. A permutation graph is the intersection graph of a family of line segments whose endpoints lie on two distinct parallel lines; vertices are adjacent if and only if their corresponding line segments cross. For the alternative non-intersection model, let 1r be a permutation of {1,2, ... ,n}, and consider the list [1r(l), 1r(2), ... 1r(n)]. Let G = (V, E) have vertices V = { v1 v2 ... vn} with vivj E E whenever i < j and i is to the right of j in the list; then G is a permutation graph. A comparability graph is a graph G = (V, E) in which E may be oriented so that for x, y, z E V if x --+ y and y --+ z, then x --+ z; that is E has a transitive orientation. A cocomparability graph is a graph whose complement's edges have a transitive orientation. 5.2 Characterizations Originally, valuation bigraphs were conceived in a more general sense in order to model simultaneous rankings, or quantifiable judgments, placed on a collection F of (not necessarily distinct) sets. If the funCtions h, ... fk represent the quantifiable criteria placed on F = {X1 ... Xk} such that fi : Xi --+ JR, and we wish to examine the rankings under the given criteria with respect to 85

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some tolerance E > 0, it may make sense to examine the k-partite graph G = (:F, E) with uv E E {::} lfi(u) -fi(v)l ::; E, where u E Xi, v E Xi. We call a graph having such a representation a valuation k-graph, and the specialization to :F = {X, Y} are the valuation bigraphs. The next result clearly specializes to valuation bigraphs. Theorem 5.1 A graph G is a valuation k-graph if and only if it is a unit interval k-graph. Proof: Let G = (X1 ... Xk, E) be a valuation k-graph with valuation functions fi : Xi --+ IR, for i = 1, ... k. Create an interval I(v) = [l(v), r(v)] for each vertex v of G as follows: put I(v) = [fi(v) fi(v) + for v E Xi, and say I ( v) belongs to interval class 'Ii. Note that if two vertices are in the same partite set, then they are not adjacent by definition. So, with x E Xi and y E Xi we show that xy E E if and only if I(x) n I(y) =/=-0. We have lfi(x) -fi(y) I ::; 1 {::} -1::; fi(x) -fi(y) ::; 1 {::} 1 1 -1::; fi(x) + 2 -fi(Y)-2 ::; 1 {::} -1::; r(x)-r(y) ::; 1 {::} lr(x) r(y) I ::; 1 {::} I(x) n I(y) =/=-0, where the last equivalence follows since all intervals have length 1. Thus, if G is a valuation k-graph, then G is a unit interval k-graph. Conversely, suppose that G = (X1 ... Xk, E) is a unit interval k-graph with interval representation :F = { 'Ii, ... 'Ik}, where I I ( v) I = 1 for each I ( v) E :F, and 86

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assume all intervals are contained in JR+. For I(v) E Ii, put fi(v) = l(v) + replace fi with fi if I(v) E Ij. Suppose xy E E with x E Xi, y E Xj; i.e., I(x) n I(y) # 0 and I(x) E Ii, I(x) E Ij. Assume without loss of generality that l(x) :::; l(y); then r(y) l(x) :::; 2. We have 0 :::; r(y)-l(x) :::; 2 {::} 1 1 -1 :::; r(y)2 -l(x)2 :::; 1 {::} -1:::; l(y) + (l(x) + :::; 1, hence -1:::; fi(y)-fi(x):::; 1 and lfi(x) -fi(y)l:::; 1. We have defined valuation functions showing G is a valuation k-graph. The following corollary to Theorem 5.1 is what will be used later, since the results of this chapter deal mainly with UIBGs. Although Corollary 5.2 and Theorem 5.1 may not be very surprising at this moment, it will be shown that the valuation representation is equivalent to many other representations, in particular, and perhaps more surprisingly, proper interval bigraphs turn out to be equivalent to valuation bigraphs. Corollary 5.2 A graph G is a valuation bigraph if and only if G a unit interval bigraph. The next result also specializes to valuation bigraphs. This result suggests that, in the general setting, a result from the perspective of partially ordered sets may be obtainable. For now, in the bipartite case, this result will serve as a step toward a characterization in terms of permutation graphs. 87

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Theorem 5.3 Let G = (XI, ... Xk, E) be a valuation k-graph. Then G = (XI, ... Xk, E) has a transitive orientation; that is, G is a cocomparability graph. Proof: We adorn vertices with sub-and superscripts to enumerate them and discern to which partite set they belong. Let Xi = {xi, ... xfxi1 } and fi : Xi --+ R(i = 1, ... k). Suppose that the vertices have been indexed so that fi(xi) for each i, and set the tolerance E = 1. Any pair of vertices belonging to the same partite set are adjacent in the complement of G; as are any pair x, y belonging to distinct partite sets if and only if lfi(x)-fi(y)l > 1, which happens if and only if fi(x) > fi(y) + 1 or fi (y) > fi ( x) + 1. Orient edges of G as follows: i i "f Xr --+ X8 1 r < s, We show that this gives a transitive orientation of the edges of G by considering the following cases. Case 1: If --+ xt and --+ then a < b and b < c. Hence, a < c and Case 2: If --+ xt and --+ xt then fi(x{) > + 1 and fi(xt). Thus, > + 1 and E E with x{ Case 3: If --+ and --+ xt then 2: > + 1. Thus, E E and x{. Case 4: Suppose --+ and --+ then > + 1 and > + 1. Therefore, > + 2 and a< c giving --+ as desired. 88

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Case 5: Consider xi and xi--+ Then fi(xi) > + 1 and > fi(xi) + 1; so, > + 2 which gives --+ as desired. With 1 :S i, j, l :S k arbitrary this covers all possible cases and so G is a cocomparability graph. Corollary 5.4 A valuation bigraph is a cocompar.ability graph. As we alluded to above, Corollary 5.4 and the next result by Pneuli, Lempel and Even [11] show that valuation bigraphs, and hence unit interval bigraphs, are permutation graphs. Theorem 5.5 (Pneuli, Lempel, Even, 1972, [11]) A graph is a permutation graph if and only if it and its complement have a transitive orientation. Theorem 5.6 If G = (X, Y, E) is a valuation bigraph, then G is a bipartite permutation graph. Proof: Assigning x --+ y for x E X, y E Y, xy E E gives a transitive orientation of G. This together with Theorem 5.5 and Theorem 5.4 gives the result. The next result, while shedding more light on the structure of valuation bigraphs, also gives another way of showing that a valuation bigraph (or a unit interval bigraph) is a bipartite permutation graph. First, we need the following definition and a corresponding theorem, Theorem 5. 7. For a bipartite graph G = (X, Y, E), there is a strong ordering on X U Y if there is an irreflexive binary relation < such that if x, x' E X, y, y' E Y, xy, x'y' E E, x < x', and 89

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y' < y, then xy', x' y E E. Here is one way to visualize this property. Order all members of X on one of two parallel lines with respect to < and the members of Y on the other line, also in accord with <. If xy E E, then let there be a line joining x and y. Whenever two lines cross, the four vertices determining those lines form a biclique. This property was introduced in [5] to characterize bipartite permutation graphs. We record this result for reference. Theorem 5. 7 (Spinrad, Brandstadt, Stewart, 1987, [5]) A bipartite graph G = (X, Y, E) is a permutation graph if and only if there is a strong ordering on XUY. Theorem 5.8 Let G = (X, Y, E) be a unit interval bigraph; then G has a strong ordering. Proof: Consider xy, x'y' E E with l(x) :::; l(x') and l(y') :::; l(y), and of course r(x):::; r(x') and r(y'):::; r(y). Since I(x) ni(y) "I 0 and I(x') ni(y') "I 0 and all intervals are unit length, either r(y') l(x) or l(y') :::; r(x); either way I(x) ni(y') "I 0, and xy' E E. Similarly, either r(y) l(x') or l(y) :::; r(x'); both possibilities give I(x') n I(y) -1 0 and x'y E E. Thus, if we index X= {xi} and Y = {Yi} so that l(x1) :::; l(x2) :::; :::; l(x1x1) and l(YI) :::; l(y2) :::; :::; l(YIYI), we have a strong order < where xi < Xj if i < j and Ya < Yb if a < b. Combining Theorem 5. 7 and Theorem 5.8, we have another way to see that a valuation bigraph is a bipartite permutation graph. Later we give a characteri zation for proper interval bigraphs in terms of a total vertex ordering which we show to be equivalent to a strong ordering. 90

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Figure 5.2: Forbidden unit interval bigraphs, include Ck, k ;::: 6 It is a well-known result of Gallai's [14] that cocomparability graphs are asteroidal triple-free, hence we may obtain the next result, originally proved directly by the author in [7] (and in [6] for UIBGs). Corollary 5.9 If G is a unit interval k-graph, then G is asteroidal triple-free. If we examine Gallai's list of forbidden subgraphs in [14] modulo all that are not bipartite, we have a list of forbidden sub graphs for bipartite cocomparability graphs, bipartite asteroidal triple-free graphs, bipartite permutation graphs, and thus unit interval bigraphs. The list is exhibited below in Figure 5.2. More can be said however and we postpone recording these results formally until Theorem 5.18. The following discussion will put our results in the context of (0,1)-matrices. A bipartite graph G = (X, Y, E) is X -consecutive if X can be ordered so that for each y E Y, N(y) is consecutively ordered; a Y -consecutive bipartite graph is defined similarly. Our next result will be used in the theorem that follows it, but in and of itself, it shows that UIBGs, and all the classes to which they are equivalent, are simultaneously X-and Yconsecutive bigraphs. The converse of this is not true, however, as G1 and G2 of Figure 5.2 are both X-and Y consecutive, but they are not UIBGs. 91

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Recall that to any bipartite graph G = ({xi},{Yj},E), with l{xi}l = m and I{Yi}l = n, there corresponds an m x n(0, 1)-matrix A( G) = (aij), with aij = 1 if and only if XiYj E E. We call A(G) the reduced adjacency matrix of G. If G = (X, Y, E) is X-consecutive, then labeling the rows of A( G) with X (which we conventionally do throughout this thesis) gives a (0,1)-matrix with the consecutive 1 's property for columns; that is, the rows of A( G) can be permuted so that the 1 's in all columns appear consecutively. Given the preceding observation, the next result of Tucker is a structural characterization for the consecutive 1's property for columns in a (0,1)-matrix. Theorem 5.10 (Tucker, 1972, [45]) A bipartite graph G = (X, Y, E) is Xconsecutive if and only if it has no asteroidal triple contained in X. A more recent result of West [47] characterizes the unit interval bigraphs in terms of a structural property of their reduced adjacency matrices that is stronger than having the consecutive 1's property for both rows and columns simultaneously. We discuss this result below, putting our results in a more exact context with regards to (0,1)-matrices. One of our next main results shows that if a bipartite graph has a strong ordering, then it is a unit interval bigraph. To this end, we first prove the follow ing lemma that also shows any bipartite permutation graph is simultaneously an X-and Y -consecutive bigraph. Lemma 5.11 Let G = (X, Y, E) have a strong ordering, and suppose that X and Y are indexed in accord with the strong ordering; that is, Xa < Xb if a < b and Yc < Yd if c < d. If XiYa., XiY/3 E E with a < /3, then XiYj E E for all j 92

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satisfying a< j < /3. Moreover, YjXa, YiXfJ E E, for a < /3, imply YjXi E E for i with a < i < b. Proof: An isolated vertex is of no consequence to the strong ordering, so we may either assume that there are no isolated vertices, or that they have largest indexes in the order. Consider XiYa, XiYfJ E E with a < /3; assume that XiYj E for some j with a< j < /3. Then there must be some index l for which XlYi E E since Yi is not isolated. We have two possibilities: l < i, or i < l. In the former case we have xl < Xi, Ya < Yi, XiYa, XlYi E E implies XiYj E E (and x1ya E E) by the strong ordering. In the latter case we have xi < xl, Yi < YfJ, xlyj, XiYfJ E E forcing XiYi, x1yfJ E j3 under the strong ordering. Either way we have XiYi E implies Yi is isolated, a contradiction. Thus, XiYj E E for all j satisfying a < j < /3. We may also make the same conclusion with the roles of x and y interchanged. More can be said, however. Similar to the way a (0, 1)-matrix represents a bipartite graph, any digraph can be represented as such with vi -+ Vj if and only if entry ( i, j) = 1; the (0, 1) matrix that represents a digraph in this way is necessarily square. In [10] and [41] interval digraphs and unit interval digraphs, respectively, have been characterized in terms of their adjacency matrices. Thus, modulo the addition of rows or columns of zeros in case G = (X, Y, E) has unequal-sized partite sets, these characterizations apply to IBGs and their cor responding subclasses. A (0, 1)-matrix has a monotone consecutive arrangement if its rows and columns can be permuted independently so that each zero can be labeled R or C in such a way that every entry below and to the left of a C is 93

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a C and every entry above and to the right of an R is an R. San and Sen in [41] proved that a digraph is a unit interval digraph if and only if its adjacency matrix has an MCA. In [47] West proved the same result using the interval bigraph model which allowed for a shorter induction proof than that given in [41). Here we give a result that ties the MCA property into this exposition via permutation graphs. We will use the following lemma from [41]. In a (0,1)-matrix, we use ai to indicate the first column in which a 1 appears in row i; similarly, f3i denotes the last column in which a 1 appears in row i. Lemma 5.12 (Sanyal, Sen, 1994 [41]) A {0,1}-matrix with n nonzero columns has a monotone consecutive arrangement if and only if its rows and columns may be permuted independently so that the 1 's appear consecutively in each row and { ai}i=1 and {f3i}i=1 form nondecreasing sequences. Lemma 5.13 If G = (X, Y, E) is a bipartite graph whose vertices may be labeled to exhibit a strong ordering, then A( G) has a monotone consecutive arrangement. Proof: Let X = {xi}, and Y= {Yi} be indexed in accord with the strong ordering; i.e., X1 < X2 < < Xm, Y1 < < Yn, and XpYs, XqYr E E, with p < q, r < s implies XpYr, XqYs E E. Permute rows and columns of A(G) = (ai,j) so that row i corresponds to xi and column j corresponds to Yi A(G) ism= lXI x IYI = n. The following argument stems from the fact that in A( G) the strong ordering forbids [ and as submatrices of A( G) = (ai,j), where E {0, 1}. Lemma 5.11 gives that the 1's in each row appear consecutively. Now we consider rows i and k with i < k. If we have ak < ai, then we 94

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have a submatrix of the form [ J contradiction. So ai ak whenever i < k and { ai} forms a nondecreasing sequence. Again, with i < k, assume f3k < f3i In this case, we have a submatrix of the form [ J contradiction. So we have f3i f3k whenever i < k, and {f3i} forms a nondecreasing sequence. Therefore, by Lemma 5.12, we conclude that A(G) has a MCA. Theorem 5.14 A bipartite graph has a strong ordering if and only if its reduced adjacency matrix has a monotone consecutive arrangement. Proof: One direction has been proved. For the other assume G =(X, Y, E) has an m x n reduced adjacency matrix A( G) with rows and columns permuted so that it exhibits a monotone consecutive arrangement. Label the vertices X = {x1, x2, ... Xm} and Y = {y1 y2 ... Yn} so that row i corresponds to xi and column j corresponds to Yj Consider a< band c < d with XaYd, XbYc E E. If XaYc E, then entry (a, c) = 0 and cannot be labeled R or C since entry (b, c) = 1 and is below it, and entry (a, d) = 1 and is to the right of it. Thus, (a, c) = 1; i.e., XaYc E E. If XbYd E, then entry (b, d) = 0 and cannot be labeled R or C since entry (b, c) = 1 and is to the left of it, and entry (a, d) = 1 and is above it. Thus, entry (b, d) = 1; i.e., XbYd E E. We have shown that the ordering of rows and columns that yield an MCA give a strong ordering in the vertices of the corresponding bipartite graph. Now we construct a unit interval bigraph representation for a bipartite graph with a strong ordering. 95

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Theorem 5.15 IJG =(X, Y, E) has a strong ordering, then G is a unit interval bigraph. Proof: Let G = (X, Y, E) have a strong ordering and vertices indexed so that XI < X2 < < Xm and YI < Y2 < < Yn, and m 2: 2, n 2: 2 so that degenerate cases are avoided. Assuming that G is connected, which poses no loss of generality since we can argue for each component separately, we will create a unit interval representation with all interval endpoints distinct, and {l(xi)} and {l(yi)} forming increasing sequences. We will use ai to denote the smallest index among all members of Y with XiYa; E E; while f3i denotes the largest index among all members of Y with XiY/3; E E. Claim 1: XIYI E E and XmYn E E. Since G is connected and hence contains no isolated vertices, there must be some Yi with XIYi E E, and taking j as small as possible, i.e., ai = j, we see that if a I = 1, we are finished. So assume a I > 1. Similarly, YI cannot be isolated; so it must be adjacent to some Xi, i > 1. We have XI < Xi, YI < Yi, XIYi E E, XiYI E E which by the strong ordering gives XIYI E E (and XiYi E E) and a1 = 1. Essentially the same argument applies to Xm and Yn Neither can be isolated, so taking j as large as possible we have XmYj E E and XiYn E E for some i < m. As Xi < Xm, Yi < Yn, XiYn E E, XmYj E E we have XmYn E E by the strong ordering. For the purposes of the next claim, let A = {x1, ... Xi-1 Y1, ... Yf3;_J, B = {xi, ... ,Xi-l,Y/3;-l+l,,Ym}, C = {xi, ... ,Xm,Yl,,Ya;-I}, and D = {xi,, Xm, Ya;, Yn} 96

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Claim 2: For i = 1, ... m, ai :::; f3i-l Assume ai > f3i-l and XaYb E E, with 1 :::; a < i -1 and f3i-l + 1 :::; b :::; n, where a =f. i-1 by definition of f3i-l Then as Xa < Xi-1, Yf3;_1 < Yb, XaYb E E, Xi-1Yf3;_1 E E we have Xi-lYb E E by the strong ordering, a contradiction to definition of f3i-l On the other hand if XcYd E E with i < c :::; m and 1 :::; d :::; ai -1, where i =f. c by definition of ai, we have Xi < Xc, Yd < Ya;, XiYa; E E, XcYd E E forcing XiYd E E under the strong ordering. This contradicts the definition of ai. Thus En B = 0 and En C = 0, implying that all edges consist of pairs from A or pairs from D, exclusively. Therefore G is not connected whenever ai > f3i-l, fori> 2. Claim 3: ai-l :::; ai and f3i-l :::; f3i, for 1 :::; i :::; m -1; that is, { ai} and {f3i} form nondecreasing sequences. Assume ai < ai-l Then Xi-l < Xi, Ya; < Ya;-1) Xi-lYa;_1 E E, XiYa; E E force Xi-lYa; E E under the strong ordering, in contradiction to the definition of ai-l since ai < ai-l If f3i < f3i-l, then we have Xi-1 < Xi, Yf3; < Yf3i-ll and xi-lYf3;_1 E E, XiYf3; E E. But then the strong ordering forces XiYf3i-1l in contradiction to the definition of f3i since f3i < Therefore { ai} and {f3i} form nondecreasing sequences. We create unit length intervals, specifying only the left endpoints (since r(v) = l(v) + 1 for all vEX U Y), beginning with i = 1 and iteratively adding the Xi-interval together with intervals for all its neighbors in Y at each step. Put J(x1 ) = [0, 1]. By Claim 1 we know x 1y1 E E and by Lemma 5.11, we know x 1yi E E for 1 = a1 :::; j :::; f3i; so let be an increasing sequence 97

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in the open interval (0, 1). Now we give the procedure for i > 1. From the strong ordering, Claim 2, and Lemma 5.11 we know XiY/3i-l E E so necessarily l(xi) < r(yf3i_J; our assumptions call for l(xi_ 1 ) < l(xi); and by definition of ai and Claim 2, l(xi) > r(Ya;1). So let f-ti = max{l(xi-1), r(Ya;-1)} and choose l(xi) E (p,i, r(yaJ) distinct from all endpoints that happen to lie in this interval. If f3i = f3i_1 then ally-intervals have been created. For the case f3i > f3i-1, we make {l(yf3i-I+1), ... ,l(yf3J} an increasing sequence in the interval (!'i,r(xi)), where 'Yi = max{l(yf3i-1), r(xi_1 )}. The choice for 'Y stems from the fact that we must make l(y/3i-I+ 1 ) > l(yf3i-l), and by definition of f3i-1, Xi-1Yk "/k ;::: r(Yk1 ) ;::: r(xi) giving I(xi) n I(yj) = 0, a contradiction. Therefore, I(xi) n I(yj) -=/= 0 if and only if XiYi E E. Interval bigraphs have been characterized recently by Hell and Huang [24] :via a total ordering on their vertices that is precisely part (i) of the ordering 98

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below. See also Theorem 3.6. The ordering below turns out to characterize proper interval bigraphs, and hence, in conjunction with the result of Hell and Huang, gives them a precise relationship to interval bigraphs. In general, the next result is in line with those regarding chordal graphs in [27], and in a similar fashion we give a forced adjacency diagram after the proof that depicts the ordering. Theorem 5.16 If G = (X, Y, E) on n vertices is a proper interval bigraph, then X U Y = { can be indexed so that the following hold: (i) For any indexes a < b < c, if Va, Vb belong to the same partition, and VaVc E E, then VbVc E E; (ii) For any indexes a < b < c, if vb, Vc belong to the same partition, and VaVc E E, then VaVb E E. Proof: Let {I(v)}vEXUY be a proper interval representation for G. Index vertices so that l(v1 ) < l(v2 ) l(vn) Then for any a < b < c we have l(va) l(vb) l(vc)and since the representation is proper r(va) r(vb) r(vc) also. Thus if I(va) n I(vc) -1-0, then r(va) ;:::: l(vc) and r(vb) ;:::: l(vc) So, if Va Vc E E and Va belongs to the same partition as Vb, then I ( Vb) n I ( vc) -1-0 and vbvc E E, giving (i). If on the other hand VaVc E E and vb is in the same partition as Vc, then with l(vb) l(vc) r(va) r(vb), we have I(va)ni(vb) -1-0 and so VaVb E E, giving (ii). Figure 5.3 is an illustration representing the adjacencies that are forced under the ordering in Theorem 5.16. 99

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(i) (ii) Figure 5.3: Forced adjacency diagram for Theorem 5.16: dashed edges are forced To illustrate the distinction between satisfying part (i) only and satisfying both, we present Figure 5.4 in which the vertices of an interval bigraph that is not a proper interval bigraph are ordered in accord with part (i), but cannot be ordered in accord with both (i) and (ii) simultaneously. We see that 2 < 4 < 5 and v4 v5 belong to the same partite set, but v2 and v4 are not adjacent in violation of (ii). Symmetry and exhaustion of cases shows that no ordering can be given to the graph in Figure 5.4 that satisfies both (i) and (ii). We will show that the ordering given in Theorem 5.16 is equivalent to a strong ordering in the sense that a bipartite graph G = (X, Y, E) has a strong ordering if and only if XU Y can be ordered as in Theorem 5.16. We prove one direction of this next; the other direction will follow after all of the results in this section are put together. 4 3 1 2 5 6 7 Figure 5.4: The labeling satisfies (i), but not (i) and (ii) of Theorem 5.16 100

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Theorem 5.17 Let G = (X, Y, E) be a bipartite graph. If XU Y can be indexed in accord with the order given in Theorem 5.16, then G has a strong ordering. Proof: Let < denote the ordering in Theorem 5.16; that is, u < v if the index given by the ordering on u is less than that for v. Take any pair of edges xy E E, x'y' E E with x < x', y' < y, x, x' E X, y, y' E Y. We consider six possibilities ranging over the possible order of x, x', y, y'. Case 1: x < x' < y' < y. Then we have x < x' < y and xy E E implies xy' E E by (i}; x < y' < y and xy E E implies xy' E E by (ii). Case 2: x < y' < x' < y. Observe x < y' < y and xy E E implies xy' E E by (ii); x < x' < y and xy E E implies x'y E E. Case 3: x < y' < y < x'. By (i), y' < y < x' and x'y' E E imply x'y E E; while (ii), x < y' < y and xy E E gives xy' E E. Case 4: y' < y < x < x'. Under (i), y' < y < x' and x'y' E Ewe have x'y E E; (ii), y' < x < x' and x'y' E E give xy' E E. Case 5: y' < x < x' < y. Observe y' < x < x' and x'y' E E; so under (ii) we have xy' E E. With x < x' < y and xy E E, we use (i) to obtain x'y E E. Case 6: y' < x < y < x'. Here we have y' < y < x' and x'y' E E enabling us to use (i) and get x'y E E; while y' < x < x' and x'y' E E allows us to use (ii) and get xy' E E. We have shown that x, x' EX, y, y' E Y, x < x', y' < y, xy, x'y' E E imply x'y, xy' E E for all possibilities. Therefore, the order given in Theorem 5.16, when restricted to X and toY, gives a strong order on XU Y. When all of the above is put together in the appropriate way we have the 101

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following result giving the equivalence of valuation bigraphs, unit interval hi graphs, proper interval bigraphs, asteroidal triple-free bipartite graphs, and bipartite permutation graphs, and a new ordering characterization for the afore mentioned. We note that the equivalence of (a) and (b) were first found (stated in the parlance of, and via the perspective of, indifference digraphs) by Sanyal and Sen in [41], and then by West (stated in the bigraph parlance) in [47]. The equivalence of (a), (b), (g), (h), and (i) were found independently by Hell and Huang [24]. Furthermore, the aforementioned results by Hell, Huang, West, Sanyal, and Sen allow us to extend Theorem 5.18 and we do so below. Theorem 5.18 For a bipartite graph G = (X, Y, E), the following are equiva lent: (a)G is a unit interval bigraph; (b) G is a proper interval bigraph; (c) With XU Y = {vi}, we can index so that for indexes a< b < c: (i) Va, vb in same partition, and VaVc E E imply vbvc E E; (ii) vb, Vc in the same partition, and VaVc E E imply VaVb E E; (d) G has a strong ordering; (e) G is a permutation graph; (f) G is a valuation bigraph; (g) G and G are both transitively orientable; (h) G has no induced subgraph isomorphic to any in Figure 5.2. (i) G is asteroidal triple-free. (j) The reduced adjacency matrix of G has a monotone consecutive arrangement. 102

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Proof: (a) ====? (b) is obvious, (b) ====? (c) is Theorem 5.I6. (c) ====? (d) is Theorem 5.I7. (d) ====? (a) is Theorem 5.I5. (d)<=? (e) was proved by Branstadt, Spinrad, and Stewart in [5]. (a)<=? (f) is Corollary 5.2. (e)<=? (g) has been proved by Pneuli, Lempel, and Even, see Theorem 5.5. (g) <=? (h) follows from Gallai [I4], as does (h) <=? (i) given that G is bipartite and thus contains no odd cycles. (d) <=? (j) is given by Theorem 5.I4. A circular arc graph is the intersection graph of arcs of a circle, and a proper circular arc graph is a circular arc graph whose representation can be made so that no arc properly contains another. Hell and Huang have proved the following result independently. Theorem 5.19 (Hell, Huang, 2003, [24]) For a bipartite graph G, the following are equivalent: (a) G is a proper interval bigraph; {b) G is a proper circular arc graph; (c) G is asteroidal triple-free; {d) G is a permutation graph; i.e., both G and G are comparability graphs; (e) G is a comparability graph; {f) G does not contain any of the graphs of Figure 5.2 as an induced subgraph. A (0, I)-matrix has a zero partition if, after a sequence of independent row and column permutations, the zeros may be labeled R or C such that below any C is a C and to the right of any R is an R. In [30], the (0, I)-matrices with a zero partition that also have a monotone consecutive arrangement are characterized. I03

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The perspective given by bipartite graphs allows this result to be extended. Namely, Theorem 5.18 gives a structure theorem for the monotone consecutive arrangement property. We will need the following collection of matrices, where Ak is a k x k matrix with k 2: 3. [1110] [1100] F1 = 0 1 1 1 F2 = 11 1 1 0010 0110 [11 0 OJ 1010 1001 ,F3 = 1 0 ... 1 1 1 ... 0 ... 0 0 1 1 This collection corresponds to the graphs of Figure 5.2 with Fi the reduced adjacency matrix for Gi, and Ak the reduced adjacency matrix for a cycle of length 2k, k 2: 3. Hence, we have the following corollary to Theorem 5.18 and also the results of West, Sanyal, and Sen. Corollary 5.20 A (0, 1)-matrix M has a monotone consecutive arrangement if and only if M has no submatrix from the set { Ak, F1 F2 F3 } or their transposes. Theorem 5.18 can be extended further if we incorporate results of Golumbic, Lowenstein, U rutia, and Trenk on the following classes of graphs. A graph G = (V, E) is a tolerance graph if there is an interval I ( v) assigned to each vertex v E V and a function t : V --+ JR+ such that uv E E if and only if JI(u) n J(v)J 2: min{t(u), t(v)}. If t(v) JI(v)J for each v E V, then G is a bounded tolerance graph. A parallelogram graph is the intersection graph of parallelograms P in which 2 sides of each P E P lie on distinct parallel lines. If we generalize P to be a collection of trapezoids, then G is a trapezoid graph. 104

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Let C be a collection of continuous curves which lie between two distinct parallel lines L1 L2 such that no line parallel to L1 or L2 intersects any C E C twice. If G is the intersection graph of C, G is a function graph, and we regard each C E C as a function. We discuss function graphs in relation to interval k-graphs in the next section. Define a ribbon to be a region bounded by two curves of C and let n be a collection of such ribbons. If G is the intersection graph of n, then G is a ribbon graph. Clearly, a permutation graph is a parallelogram graph, a trapezoid graph and a function graph; parallelogram graphs, function graphs, and trapezoid graphs are ribbon graphs. Now we put West's, Hell and Huang's, Sanyal and Sen's, ours, and the afore mentioned work on tolerance, function, ribbon, and trapezoid graphs together to obtain the following list of equivalences for bipartite graphs and (0,1)-matrices. Theorem 5.21 For a bipartite graph G = (X, Y, E), the following are equiva lent: (a) G is a unit interval bigraph; (b) G is a proper interval bigraph; (c) With XU Y = {vi}, we can index so that for indexes a< b < c: (i) Va, vb in same partition, and VaVc E E imply vbvc E E; (ii) Vb, Vc in the same partition, and VaVc E E imply VaVb E E; (d) G has a strong ordering; (e) G is a permutation graph; (f) G is a valuation bigraph; (g) G and G are both transitively orientable; (h) G has no induced subgraph isomorphic to any in Figure 5.2. 105

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(i) G is asteroidal triple-free. (j) The reduced adjacency matrix for G has a monotone consecutive arrange ment. (k) G is a proper circular arc graph. (l) The reduced adjacency matrix for G has no submatrix from the set {Ak, F1 F2, F3 } or their transposes. (m) G is a ribbon graph. (n) G is a bounded tolerance graph. ( o) G is a trapezoid graph. (p) G is a function graph. Proof: The equivalences (a) through (l) have been discussed. (m) {:::} (p) is proved in [18]. (n) {:::} (o) is observed in [19]. It is interesting that such a wide range of representations are available for unit interval bigraphs. 5.3 Proper IkGs For the author, the wealth of properties of UIBGs has stymied the investigation of unit IkGs, for k > 2. Their characterization is still an open problem. We reiterate that UikGs are asteroidal triple-free, for any k, but although this characterizes them fork= 2, it does not fork > 2. In Figure 5.5, we give some of the non-bipartite asteroidal triple-free graphs that we know to be forbidden as UikGs. The list given is not complete. A proper interval k-graph is an interval k-graph with a representation in which no interval properly contains another. Observe that F of Figure 5.5 is 106

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) \1 u v Figure 5.5: Forbidden UikGs that are asteroidal triple-free. not a proper IkG because, by some symmetry aruments and exhaustion of cases, it becomes clear that in any interval representation, I(x) must be contained in I(z). It is not known whether the class of UikGs is identical to the class of proper IkGs, but we suspect that this is the case. One trait proper IkGs share with UikGs is that they are asteroidal triple-free. Theorem 5.22 If G is a proper !kG, then G has no asteroidal triple in any induced subgraph. Proof: By way of contradiction, assume G = (V, E) is a proper IkG and that {x,y,z} C Vis an AT of G. We may assume that in the interval rep resentation for G, l(x) l(y) ::; l(z), and of course r(x) r(y) r(z). Let (x, u1 u2 ... Um, z) be a path from x to z. Since I(y) can neither properly con107

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tain nor be properly contained in any other interval, there is some pair ui, ui+l for i E {1, ... m-1} with (I( ui) n I( ui+1)) n I(y) nonempty. Both ui and ui+l cannot both be contained in the same interval class since they are adjacent, and so one of ui, ui+l is in a different interval class than, and is hence adjacent to, y, contradiction. Our final result of this chapter gives a containment relationship between function graphs and proper IkGs. The theory for function graphs is a little more rich than that for IkGs (they were first studied in 1984 [18]) and hence we hope that this result proves useful or brings more interest to IkGs. Function graphs are cocomparability graphs and hence asteroidal triple-free, see [18]. To see that a function graph is a cocomparability graph, observe that vertices u, v are not adjacent if and only if their corresponding functions fu, fv do not cross anywhere in the representation and hence using the rule u -+ v if for every domain value x, fu(x) > fv(x) gives a transitive orientation of the complement of any function graph. Function graphs are also studied and placed among some of the well-known classes of intersection and tolerance graphs in the new book Tolerance Graphs by Golumbic and Trenk, [19]. Theorem 5.23 Let G be a proper interval k-graph, k 2: 2. Then G is a function graph. Proof: Let G = (V, E) be a proper interval k-graph and I1,I2 ... ,Ik the interval classes for G. Assume that all endpoints of intervals are distinct in the representation. Let {v1 ... ,vn} = V and indexing correspond to l(v1 ) < 108

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l(v2 ) < < l(vn) in the interval representation; since the representation is proper, r(vi):::; r(v2):::; :::; r(vn) as well. Take k + 1 parallel copies of the real line L0 L 1 ... Lk, and identify the last k lines with the interval classes, respectively. Identifying the roles of L0 and Lk, for each vertex v E V with I(v) E Ii, put a point Pi(v) = l(v) on Li, a point Pi+1(v) = r(v) on Li+l, and a point on all the other lines equal to l ( v). Connecting points Pi ( v) on the lines Li for 1 :::; i :::; k + 1, and calling the connected points fv, we claim, gives a function representation for G. Notice that the construction g1ves functions with vertical components, one positive-slope component, and one negative-slope component. For v with I(v) E Ii the negative-slope component occurs between Li and Li+l, the positive slope component occurs between Li+l and Li+2 and the rest of fv is vertical (subscripts modulo k). If uv E E, then I(u) n I(v) =f-0 and I(u), I(v) belong to distinct interval classes. We can assume l(u) < l(v), and of course r(u) < r(v); so, either the negative-and positive-slope components of fu, fv cross, or the vertical component of one of fu, fv crosses both the positiveand negative-slope components of the other. Thus, fu n fv =f-0. If on the other hand uv E, then either I(u),I(v) both belong to, say Ii, and, by the construction and since the interval representation is proper, fu n fv = 0 because their negativeand positive-slope components lie between the same lines: Li, Li+ 1 and Li+ 1 Li+2 respectively. Clearly, if uv E because I(u) n I(v) =f-0, then fun fv = 0. Therefore, {fv}vEV is a function representation for G. Figure 5.6 illustrates Theorem 5.23. Note that Theorem 5.23 gives another 109

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proof of Theorem 5.22, by the remarks preceding the theorem. Also, when k = 2, that is, G = (X, Y, E) is a proper interval bigraph, observe that the construction gives two copies of a permutation representation of G. And when k = 2, the construction reverses: take one of the permutation representations, project the negative-slope lines onto a copy of JR, call this the interval class for X, project the positive-slope lines onto a copy of lR and call this the interval class for Y. It is easy to check that this construction gives a proper interval bigraph representation for G. Hence, we have another proof of Theorem 5.21 (e). Figure 5.7 shows that the converse of Theorem 5.23 is not true, and Figure 5.8 illustrates the above comments about the construction for proper interval bigraphs. Corollary 5.24 Proper IkGs are cocomparability graphs. Corollary 5.25 Unit IkGs are cocomparability graphs. Proof: A unit representation is a proper representation. Corollary 5.26 Proper IkGs are asteroidal triple-free. Corollary 5.27 Proper IBGs are precisely bipartite permutation graphs. Proof: Clearly, one direction is given by the construction in Theorem 5.23. On the other hand, given a permutation representation of a bipartite graph G, make all line segments with nonnegative slope one partite set, project them onto lR as intervals, and let them be one family of intervals. Then make all line segments with negative slope the other partite set, map them onto lR and 110

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H 1 2 3 \l\1 4 5 Lo Lz= Figure 5.6: A function representation for a proper interval 3-graph ala Theo rem 5.23. Directly below H is its proper interval k-representation, and directly below that is a function representation for H. let them be the other family of intervals. The two families of intervals form a proper interval bigraph representation for G. 111

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1 2 4 6 5 3 1 2 4 6 5 3 Figure 5. 7: A function representation for F: an interval k-graph that is not a proper IkG. 1 2 3 rn 4 5 6 4,....--;-"""'1 Figure 5.8: A function representation, and a permutation representation for a proper interval bigraph. 112

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6. Unit Probe Interval Graphs This chapter is devoted to unit probe interval graphs. The main result of this chapter is the characterization of cycle-free unit probe interval graphs by for bidden substructures. But before this characterization is presented, we explore some of the general properties of unit probe interval graphs and their relation ships with some other well-known intersection and tolerance graphs. We close with a conjecture for the forbidden subgraph characterization for bipartite unit probe interval graphs, and discuss the partial order found on the complement of a unit probe interval graph. 6.1 Background Recall that a probe interval graph (PIG) is a graph G = (V, E) with partition V = ( P, N) and a collection of intervals I of R representing the vertices such that vertices are adjacent if and only if their corresponding intervals intersect and at least one vertex belongs toP. The sets P and N are called probes and nonprobes, respectively, and the triple (I, P, N) is called a probe interval representation. To denote the interval representation of a subgraph H of G, we use I(H). If G is a PIG with the property that in some (I, P, N) each interval has the same length, then G is a unit probe interval graph (UPIG); if (I, P, N) can be made so that no interval properly contains another, then G is a proper interval graph (PPIG). In this chapter we report on UPIGs. In particular, we will show that they are cocomparability graphs via two proofs, and we will give a forbidden induced subgraph characterization for the cycle-free case. One of the 113

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ways we show UPIGs are cocomparability graphs is direct and uses the probe interval representation, the other capitalizes on a relationship between PIGs and tolerance graphs and parallelogram graphs. If the partition V = (P, N) is known or given and G = (V, E) has a probe interval representation, then G is a partitioned probe interval graph. The task of deciding whether a graph with a given partition V = (P, N) is a PIG is easier than the same task when there is no partition given. PPIGs and UPIGs are one and the same, see [19] and [31]. This result is recorded here for use later. Theorem 6.1 (Lipshteyn, [31]) Agraph is a UPIG if and only if it is a PPIG. One of the first structural results we obtained for UPIGs was the following. It places UPIGs in a class of graphs that are conducive to some nice algorithms and says basically that UPIGs can grow in only two directions, see [9]. Recall that an asteroidal triple is a set of three vertices with a path between any two that avoids the neighborhood of the third. Theorem 6.2 If a graph is a UPIG, then it has no asteroidal triple in any induced subgraph. Proof: We use the proper interval model which is equivalent to the unit model by Theorem 6.1. By way of contradiction, assume G = ( P, N, E) is a PPIG and,that {x, y, z} is an AT of G. We may assume that in the probe inter val representation for G, l(x) ::; l(y) ::; l(z), and of course r(x) ::; r(y) ::; r(z). Let (x, u1 u2 ... um, z) be a path from x to z. Since I(y) can neither prop erly contain nor be properly contained in any other interval, there is some pair ui,ui+l fori E {1, ... ,m-1} with (I(ui) ni(ui+I)) ni(y) nonempty. Both ui 114

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and ui+l cannot both be contained in N, and so one of ui, ui+l is adjacent to y, contradicting the assumption that { x, y, z} is an asteroidal triple. Therefore, the result follows. A cocomparability graph G is one in which the edges of G can be directed so that the resulting digraph is transitive. As discussed in Chapter 1, because we are concerned with orienting each edge of the complement of a graph, we may consider a partial order to be a binary relation on a set that is irrefiexive and transitive. Note that the order defined by a transitive orientation on a graph or its complement (if one exists) is vacuously antisymmetric. So, if a graph G = (V, E) is a cocomparability graph, then a transitive orientation of its complement gives a partial order on V. Thus, vertices are adjacent in G if and only if they are not comparable in the partial order; that is, with < representing the order relation, u, v E V, then uv E E if and only if u f. v and v f. u. Thus, a cocomparability graph is the incomparability graph of some partial order. The next section is devoted to showing that UPIGs are the incomparability graphs of, perhaps a useful or interesting, partial order. Aside from achieving this result, the approaches we use will show relationships between UPIGs and other intersection and tolerance graphs. 6.2 UPIGs are Cocomparability Graphs Our first proof of the main result of this section is direct and uses the structure ofthe intervals which represent the UPIG. Beyond its outright purpose, the proof shows that a UPIG is the incomparability graph of a partial order that is the interplay of a semi-order and a linear order. Recall that unit interval 115

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graphs are the incomparability graphs of semi-orders. See [40], [39], and [20] for more on semi-orders and their relationships with unit interval graphs, and [32], and [42] for more on the order theoretic properties of semi-orders. Theorem 6.3 Let G = (P, N, E) be a UPIG. Then G has a transitive orienta tion; i.e., G is a cocomparability graph. Proof: Let I= {I(v) = [l(v), r(v) = l(v) + 1] : v E PUN}, and (I, P, N) be a probe interval representation for G. We will use the left endpoints to define the transitive orientation. For any pair of vertices u, v E N, uv l(x) + 1, or (2) l(x) > l(u) + 1. Put x---+ u if (1) holds, and u---+ x if (2) holds. For vertices x, yEP, xy E E(G) if (a) l(y) > l(x) + 1, or (b) l(x) > l(y) + 1. Put x---+ y if (a) holds, andy---+ x if (b) holds. We must show this gives an orientation that is in fact transitive. We consider the situation on three vertices x, y, z with x ---+ y, y ---+ z in G, and show that x ---+ z. There are eight cases to consider depending on whether none of x, y, z are in P (1 possibility), one of x, y, z are in P (3 possibilities), two of x, y, z belong to P (3 possibilities), or x, y, z E P (1 possibility). We adorn vertices with a subscript N to denote membership inN, and a subscript of P to denote membership in P. Case 1: XN ---+ YN, YN ---+ ZN Then l(x) :::::; l(y) :::::; l(z), and so l(x) :::::; l(z) giving x and z adjacent in G, and x---+ z. Case 2: Xp ---+ YN, YN ---+ ZN Then l(y) > l(x) + 1, l(y) :::::; l(z); so l(z) 2: l(y) > l(x) + 1, hence xz E E(G), and x---+ z. Case 3: XN ---+ yp, yp ---+ ZN. Then l(y) > l(x) + 1, l(z) > l(y) + 1, and so 116

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l(z) > l(y) + 1 > l(x) + 2; this certainly gives xz r:J_ E(G), and x---+ z. Case 4: XN ---+ YN, YN ---+ Zp. Then l(x) l(y), l(z) > l(y) + 1, and certainly l(x) + 1 l(y) + 1; so l(z) > l(y) + 1 l(x) + 1 showing x and z not adjacent in G, and x ---+ z. Case 5: Xp---+ yp, YP---+ ZN. Then l(y) > l(x) + 1, l(z) > l(y) + 1, and as in Case 3 l(z) > l(y) + 1 > l(x) + 2. Thus, xz r:J_ E(G), x---+ z. Case 6: xN---+ yp, yp---+ Zp. This is the same as Case 5: l(y) > l(x) + 1, l(z) > l(y) + 1 implying l(z) > l(y) + 1 > l(x) + 2 and xz r:J_ E(G) and x---+ z. Case 7: Xp---+ YN, YN---+ zp. This is similar to Case 6: l(z) > l(y) + 1 > l(x) + 2 so xz r:J_ E( G) and x ---+ z. Case 8: Xp ---+ yp, yp ---+ Zp. Again l(y) > l(x) + 1, l(z) > l(y) + 1, giving l(z) > l(y) + 1 > l(x) + 2 and xz r:J_ E(G) and x---+ z. This covers all possibilities and shows that the orientation given to G is transitive; therefore G is a cocomparability graph. Cocomparability graphs have been shown to be asteroidal triple-free, in [17] and in [14], whence Theorem 6.2 follows as a corollary. Corollary 6.4 If G is a UPIG, then G does not contain an asteroidal triple in any induced subgraph. We present another proof of Theorem 6.3 that relies on a relationship be tween PIGs and a couple of classes of graphs, the first of which was defined originally in [17] and, like PIGs (but not in a way that is similar), generalizes in terval graphs. A graph G = (V, E) is a tolerance graph if there exists a collection 117

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at= et= Nl a b c d e ft= =l Figure 6.1: NC7 is a tolerance graph, not a bounded tolerance graph. I = {I ( v) : v E V} of closed intervals of IR and a tolerance function f : V ---+ JR+ so that, for x, y E V, xy E E if and 6nly if II(x) n I(y) I 2: min{ t(x), t(y)}. The ordered pair (I, t) is called tolerance representation of G. We need to define a few special classes of tolerance graphs. If G = (V, E) is a tolerance graph and in some tolerance representation, t( v) ::; II ( v) I for each v E V, then G is a bounded tolerance graph; if I I ( v) I = 1 for each v E V, then G is a unit tolerance graph; if in some tolerance representation t(v) = for each v E V, then G is a 50% tolerance graph. Note that in a unit tolerance graph, since no pair of intervals can overlap in more than the assigned length for the intervals, any unit tolerance graph is a bounded tolerance graph. In Figure 6.1 we give a tolerance representation for the graph NC7; the shaded rectangles attached to the intervals represent the tolerance for the corresponding vertex, while the tolerance of vertex 8 is infinity. It can be shown that t(8) > II(8) I necessarily, and hence NC7 is not a bounded tolerance graph. We will use the following result that has been observed many times unof ficially, e.g.,[l6], and first appeared in the literature in [19]. It shows PIGs are tolerance graphs, but the converse is not true. The graph in Figure 6.2, that 118

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a a 1i c b\z=r--'--=j""'' """": =j b b e e \z= =1 f : Figure 6.2: A unit tolerance graph that is not a PIG. we proved is not a PIG in Chapter 2, is a tolerance graph. In fact, that the graph in Figure 6.2 is a unit tolenirice graph as evinced by the representation, makes one wonder about the relationship, if any, between PIGs or UPIGs and unit tolerance graphs. Lemma 6.5 If G is a PIG, then G is a tolerance graph. Proof: If G = (V, E) is a PIG, then let (I, P, N) be a probe interval representation for G. To define a tolerance representation for G, we will use I and define t : V -+ JR+ by first choosing E > 0 arbitrarily small, and then { E ifvEP put t( v) = .f N. It is easy to check that the pair (I, t) is a tolerance oo 1 v E represenation for G. The proof of Lemma 6.5 shows that UPIGs are unit tolerance graphs, since no modifications of the probe interval representation are made. Corollary 6.6 If G is a UPIG, then G is a unit tolerance graph. We present another class of intersection graphs that will be used for one of the proofs that UPIGs are cocomparability graphs. It was mentioned in the 119

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chapter on unit interval k-graphs. Let L1 and L2 be two parallel lines in and P = { Px} a collection of parallelograms such that two sides of each Px E P lie on L1 and 2 The intersection graph of Pis called a parallelogram graph. Note that an interval graph is a parallelogram graph in which the slope of each side not on L1 or L2 is fixed. Figure 6.3 is an example of a parallogram graph that is not an interval graph. Consider a line L that lies between, and is parallel to L1 and 2 We can create an order << on P for some parallelogram graph G = (V, E) in which Px < < Py if for any line L, any point a E Px on L, and any point f3 E Py also on L, a < f3. Since, Px < < Py and Py < < Pz implies Px << Pz, and Pu << Pv or Pv << Pu if and only if uv :_ E, any parallelogram graph is a cocomparability graph. We believe this was first observed in [29]. Lemma 6. 7 (Langley, 1993, [29]) If G is a parallelogram graph, then G is a cocomparability graph. The following result by Langley in [29] gives, as a corollary, that UPIGs are cocomparability graphs. Theorem 6.8 (Langley, 1993, [29]) A finite graph is a bounded tolerance graph if and only if it is a finite parallelogram graph. Corollary 6.9 If a graph is a UPIG, then it is a cocomparability graph. Proof: Suppose a graph G is a UPIG. Then it is a unit tolerance graph, by Corollary 6.6, and hence it is a parallelogram graph, by Theorem 6.8. Therefore G is a cocomparability graph by Lemma 6.7. 120

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Figure 6.3: A parallelogram graph. xoy w z Figure 6.4: A parallelogram representation for a bounded tolerance graph. Theorem 6.8 is illustrated in Figure 6.4. The restriction that the graphs are finite, while assumed in general for this thesis, is emphasized because the proof relies on the ability to shift one of the lines L1 or L2 sufficiently to the left or right so that all the slopes of the sides not on L1 or L2 are negative. 6.3 Cycle-free Unit PIGs We will show that G is a cycle-free unit probe interval graph if and only if it contains no F1 F2 or Hk; for k 2: 0, of Figure 6.7 as an induced subgraph. In Figure 6.5 the darkened vertices represent probes while the white vertices 121

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represent nonprobes. The fact that a cycle-free UPIG cannot have F1 of Figure 3 as an induced subgraph, regardless of probe-nonprobe partition, shows that any cycle-free UPIG must be a forest of caterpillars. A caterpillar is a tree with a path such that every vertex has distance at most one from this path. Given a tree G, we will show that G is a partitioned UPIG, if and only if G has no induced subgraph isomorphic to any of the graphs G1 G2 G3 G4, or Gs of Figure 6.5. The graph K1 3 plays an important role in these investigations; in particular, its presence as an induced subgraph to an extent forces the probe-nonprobe partition. To be precise, we have the following two lemmas which will refer to Figure 6.6. Figure 6.5: Forbidden partitioned cycle-free UPIGs darkened vertices are probes. Figure 6.6: Labeling used in lemmas 3.1, and 3.2. Lemma 6.10 The vertex of degree 3 in any induced K1 3 of a UPIG must be a probe. 122

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Ho Figure 6.7: Forbidden PPIGs and UPIGs. Proof: Referring to Figure 6.6, assume on the contrary that w EN; then x, y, z E P since each of x, y, and z are adjacent to w. Construct I( (x, w, y) ); necessarily I(z) n I(y) = 0 and since all intervals have the same length, I(z) must intersect either I(x) or I(y), contradiction. Lemma 6.11 IJG is a UPIG, then in any induced K1 3 ofG at least two of the vertices of degree 1 must be nonprobes. Proof: Suppose by way of contradiction G = (P, N, E) is UPIG containing the graph in Figure 6.6 as an induced subgraph with x, yEP (wE Pis forced by Lemma 6.10). If we construct I( (x, w, y) ), then the requirement I(x) ni(y) = 0, and that all intervals are the same length forces I(z) to intersect either I(x) or 123

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I(y), a contradiction. Therefore, the result follows. The graphs F1 and G5 each have an asteroidal triple (AT) on the sets {a, b, c} and { x, y, z }, respectively. F1 is the smallest tree that is not a caterpillar, and it is well-known that a tree free of F1 as an induced subgraph is a caterpillar. We will capitalize on this fact below. Lemma 6.12 (Sheng, [43]) Let G = (V, E) be a graph, P V, N = V \ P, where G(N) is an independent set. If G has an AT contained in P, then G is not a probe interval graph. Lemma 6.13 If G = (P, N, E) and contains G1 G2 G3 G4, or G5 of Figure 6.5 as an induced subgraph, then G is not a UPIG. Proof: Assume G = (V, E) is a graph and V = (P, N) is given. Lemma 6.10 and Lemma 6.11 show that none of G1 G2 and G3 can be an induced subgraph. Now assume G4 is an induced subgraph of G. We may assume r(x) < l(w) and r(w) < l(y). Since I(x) n I( a) =!= 0, I(b) n I(y) =I= 0, I( a) n I(w) =I= 0, and I(w) n I(b) =/= 0, we must have either I(z) n I(a) =/= 0 or I(z) n I(b) =/= 0 since all intervals are unit length, a contradiction. Therefore, it is impossible to place I(z). Lemma 6.12 disallows G5 as an induced subgraph since there is an asteroidal triple on the set { x, y, z }. Lemma 6.14 If a tree G = (P, N, E) has no induced G1 G2 G3 G 4 or G5 of 124

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Figure 6.5, and G(N) is an independent set, then G has no induced F1 of Figure 3, and so G is a caterpillar. Proof: Assume G = (P, N, E) has F1 as an induced subgraph but has no Gi (1 i 5) of Figure 6.5. Then {a, b, c} of V(F1 ) cannot all be probes, since there is no induced G5 So, without loss of generality suppose b E N; then e must be a nonprobe since G(N) is an independent set. No induced G1 forces g of V(F1 ) to be a probe, and no G2 or G3 forces d, f E N, so a and care probes. But now we have an induced G4 contradiction. For the next result we introduce some terminology we will use for caterpil lars. If G is a caterpillar, then it has a set of vertices X with G(X) a longest path to which all vertices either belong, or are adjacent to a vertex of X. Hence the vertices of a caterpillar G = (V, E) can be partitioned V = (X, Y), where X is as in the preceding sentence and Y = V \ X. We call the set Y the pendant vertices of caterpillar G, and sometimes we say v E Y is pendant to G(X) or to a vertex of X. Note that we use the term 'pendant' with respect to G(X); the end-vertices of G(X) are pendant in the usual sense, that is, they are of degree one, but we do not call them 'pendant'. Lemma 6.15 Suppose caterpillar G = (P, N, E) is a graph with G(N) an inde pendent set. If there is a longest path P* = G(X) = (xi, x2, ... Xm) satisfying the following conditions, then G is a UPIG with respect to the given partition and with P the set of probes. 1. For each Xi EN (1 i m), there is no vertex pendant to xi; 125

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2. For each xi E P (1 ::::; i::::; m), N(xi) n Y N, where Y is the set of vertices not on P*; 3. For each Xi E P with a pendant vertex (2 < 't < m 1), Xi_1 E N or Xi+l EN. Proof: We will construct a probe interval representation for G with all intervals having the same length. For each Xi of P*, let !(xi) = [i,i + 1]. If Xi E N, it has no vertex pendant to it. Consider Xi E P; then N(xi) n P N, and Xi-1 E N or xi+l E N. If Xi-1 E N, then let the intervals for the vertices pendant to Xi be [i0.5, i + 0.5]. If xi_1 E P, then xi+l EN, so let the intervals for the vertices pendant to Xi be [i + 0.5, i + 1.5]. It is easy to check that all intervals have length 1 and uv E E if and only if I(u) n I(v) =1-0 and one of u,v E P. Theorem 6.16 Let G = (P, N, E) be a tree. G is a partitioned UPIG if and only if G ( N) is an independent set and G has no subgraph isomorphic to G1, G2, G3, G4, or Gs of Figure 6.5. Proof: If tree G = (P, N, E) is a partitioned UPIG, then G(N) is an inde pendent set by definition and has no induced G 1 G 2 G 3 G 4 or G5 by Lemma 6.13. Assume G = (P, N, E) is a tree with G(N) an independent set and with no induced G 1 G 2 G 3 G 4 or G5 of Figure 6.5. To show that G is a partitioned UPIG, it suffices to find a longest path satisfying the three conditions of Lemma 6.15. We know that G is a caterpillar by Lemma 6.14, hence we can find a vertex partition V =(X, Y), where G(X) is a longest path as in the discussion 126

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preceding Lemma 6.15. Let P* = G(X) = (x1 x2 ... Xm) Since there is no, induced G1 each xi E N has no pendant vertices, satisfying property 1. With no induced G 2 G 3 or G4 it is easy to check that property 2 is satisfied. If there were some Xi E P with Xi-l, Xi+l E P, then we would have G3 or G2 as an induced subgraph. Thus, by Lemma 6.15, G is a UPIG. Lemma 6.17 Let G be a cycle-free UPIG. Then G contains no induced subgraph isomorphic to F1 F2 or Hi, 0, of Figure 6. 1. Proof: We have shown G cannot contain F1 as an induced subgraph. Consider F2 with w1 w2 w3 each as the vertex of degree 3 of an induced K1 ,3; {w1,w2,w3 } P, by Lemma 6.10, contradicting Lemma 6.11. So, G cannot contain F2 as an induced subgraph. By Lemma 6.10, Vi, Wi of H0 must belong toP fori= 1, 2; then Y1, x2 EN so that there is no induced G3 This cannot happen since y1 and x2 are adja cent. The vertices v1,w1,e1,v2,w2 of H1 must be probes by Lemma 6.10, and since y1 E N and a1 E P is forced, b1 E N is forced. This however forces x2 E P giving an induced G3 The argument we used to show that H1 is not permitted generalizes, hence we argue next for Hk, k 2. The vertices v1, w1 v2 w2 and ei, fori= 1, ... k, must be probes, forcing y1 x2 EN. Thus,, a1 E P ==>b1 E N ==>a2 E P ==>b2 EN ==>==>bk E N, but then bk and x2 are adjacent nonprobes, contradiction. We now state and prove the characterization of cycle-free UPIGs. Theorem 6.18 Let G = (V, E) be a cycle-free graph. Then G is a UPIG if and 127

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only if G has no F1 F2 or Hi, fori 2: 0, of Figure 3 as an induced subgraph. Proof: We have proved one direction (Lemma 6.17). For the other direc tion we will create a partition V = (P, N) and an interval representation with all intervals of unit length. Assume that G has no induced F1,F2 or Hi fori 2: 0. We may assume that G is a tree since, if G is not connected we may apply the argument to each component. Furthermore, since G has no induced F1 G is a caterpillar. For some XCV with G(X) a longest path of G and each vertex ofY = V\H adjacent to some vertex of X, put Y N. Let P* = G(X) = (h1, h2, ... hm), and consider the set ordered linearly by hi < hi if i < j. For 2 ::; i ::; m -1, let each hi with a neighbor in Y be a probe, and for any two consecutive probes, say hi, hi+l, put hi_1 hi+2 E N. Since G has no induced F2, so far, there will not be three consecutive probes on P*. Since there is no induced H0 this assignment scheme so far will not produce two consecutive nonprobes. A vertex hi of P* with no pendant vertex is called reducible if hi-l and hi+l both have no neighbors in Y; i.e., hi_1 hi, and hi+l have no pendant vertices. Find all reducible vertices and make them probes. At this point a vertex that has been placed in P or N will be called a signed vertex and unsigned otherwise. A subgraph of G isomorphic to Hk( {x1 v1 w1 y1 z1 u1}) will be called 1r, and one isomorphic to Hk({ai,ei,bi,ci}) will be called aT, see Figure 6.9. Since G can have many such subgraphs, we will assume that they are ordered with respect to P*. That is, we index these subgraphs so that 1ri (Ti) is closer to h1 than 7rj (Tj) when i < j. In order to be specific when referencing vertices, 1ri will be thought of as having labels as in Hk ( { x1 v1 w1 y1 z1 u1}), but with 1 replaced with i, 128

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and similarly for Tj Note that a given vertex v may be Yi and aj simultaneously for some i and j, or bj and xi simultaneously, etc. Put hi E N and hm E N unless h2 is XI or hm-I is Yt, where 1ft is the last induced 1r of G. At this point there is no consecutive set of more than 2 unsigned vertices. To see this, suppose the consecutive set {hi,hi+l, ... ,hi+s}, for s 2 2, con sists of unsigned vertices. Then none have pendant vertices and so the set {hi+I, ... hi+s-I} consists of reducible vertices and hence belongs toP, contra diction. We call a pair of consecutive adjacent unsigned vertices an unsigned edge. The following descriptions are for some i and j, and are with reference to the ordering of P*. There are seven possibilities giving a single unsigned vertex v that is not one of an unsigned edge: (1) vis ai+l and bi, (2) vis bi and is followed by Xj, (3) vis ai and is preceded by yj, (4) vis ai and is preceded by a reducible vertex, (5) v is bi and is followed by a reducible vertex, (6) v is preceded by Yi and followed by a reducible vertex, (7) v is preceded by a reducible vertex and followed by xi For cases 2,3,6, and 7, put v E P, otherwise put v E N. See Figure 6.8. Assignments as in 1,4,5,6, and 7 will never pose any problem. The assignment in 2 is problematic only if the ai preceding v is a probe, which happens only if an assignment as in 3 occurred at ai, which happens only if HI is an induced subgraph. In the preceding sentence, replace 2 with 3 and ai with bi to get that assignment 3 is problematic only if HI is an induced subgraph. There is one possibility giving an unsigned edge uv E E: u is bi and v IS ai+l, for some pair Ti, Ti+I To deal with all possible remaining un-129

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v v li 6 bi Xj 6 6 1 2 v 4 5 6 v v 6 6 Yj ai 6 bl 3 7 Figure 6.8: Cases for an unsigned vertex v-reducible vertices are labeled r, darkened vertices have been assigned to P. Zi ui Figure 6.9: and 7ri signed vertices, suppose we have a sequence of unsigned edges of the form and ak E P, for i + 1 ::::; j ::::; i + s + 1. We have a problem if ai E P and bi+s+l E P, but this happens only if Hs+2 is an induced subgraph. Finally if h2 is unsigned, put h2 E P, and if hm-l is unsigned, put hm-l E N and hm E P, switching hm 's assignment if needed. The assignment scheme has produced a caterpillar G = (P, N, E) with G(N) an independent set and a longest path P* satisfying the requirements in Lemma 6.15, hence G is a UPIG with respect to the given partition where Pis the set of probes. 130

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6.4 Bipartite UPIGs We will reference the following figure for the fobidden subgraphs for bipartite UPIGs. X b z X b z X b z :rJ rn LIJ a y c a y c a y c G6 G7 Gg X b Z X b Z X b Z :lQ kEJ ::EJ a y c a y c a y c Figure 6.10: Forbidden bipartite UPIGs (include F1 F2 and Hi, i 2: 0). Lemma 6.19 If G = (V, E) is a bipartite UPIG, then G6 G7 G8 G9 G1o, and G11 are not induced subgraphs of G. Proof: Suppose G = (P, N, E) is a bipartite UPIG. Let Gi = ( {a, b, c, x, y, z }, Ei), as in Figure 6.10. We will use Proposition 2.1 and Lemma 6.10. By Proposition 2.1, there are no adjacent probes in any induced 4-cycle. But in each of Gi, 6 i 11, Gi -{ x, z} is isomorphic to K1 3 with y as center, and Gi-{a, c} is isomorphic to K1 3 with bas center. Thus, by Lemma 6.10, 131

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b, y E Pin each Gi giving an induced 4-cycle with adjacent probes, contradicting Proposition 2.1. We posit the following conjecture, the truth of which the author is absolutely certain, but is also certain that the proof for sufficiency is not ready for publi cation. By assuming that G is a cocomparability graph economizes only in so far as we may throw out F1 Recall that bipartite cocomparability graphs are precisely unit interval bigraphs and that the list in Figure 6.10 clearly shows that bipartite UPIGs are a subclass of unit interval bigraphs. Conjecture 6.20 A bipartite cocomparability graph is a UPIG if and only if it has no F2 Hi, i 2:: 0 (of Figure 6. 7}, G6 G7, Gs, Gg, G10 or Gn (of Figure 6.10} as an induced subgraph. Recall that Corollary 6.6 shows any UPIG. is a unit tolerance graph. The following result of Bogart, Langley, Jacobson, and McMorris gives a forbidden subgraph characterization for the cocomparability graphs that are bipartite unit (or 50%) tolerance graphs, and together with Conjecture 6.20, provided that it. is true, shows the precise relationship between bipartite UPIGs and bipartite unit tolerance graphs. Setting the truth of the conjecture aside, the following result certainly substantiates it. Theorem 6.21 (Bogart, Jacobson, Langley, McMorris, 1999, [3]) Let G be a finite bipartite cocomparability graph. Then G is a unit tolerance graph if and only if it contains no G6 G7 G8 G9 G10, or G11 of Figure 6.10 as an induced subgraph. 132

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6.5 Concluding Remarks One reason for studying graphs that are cocomparability graphs or restrict ing perspective to only those graphs that are cocomparability graphs is that this restriction allows for the investigation of the partial order that arises on the complements of the graphs under consideration. In his Ph.D. thesis and in [40] Roberts characterizes the unit interval graphs as those graphs whose complements give rise to a semi-order, a partial order developed by Scott and Suppes in [42] to model the results of Luce in [32] obtained studying preference relationships. We now define a partial order that generalizes semiorders; it is the order that comes from the complement of a UPIG. Let V be a set and j, g two bijective functions on V with f : V ---+ R, and g : V ---+ { P, N}, for x, y E V and some 6 > 0, define x < y if either f(x) > f(y) + 6 or f(x) > f(y) and g(x) = g(y) = N. Call the pair (V, <) a probe semi-order (PSO), and note that if g(x) = P for each x E V, then (V, <) is a semiorder; so PSOs generalize semi-orders the way UPIGs generalize unit interval graphs. It would be nice to see an application that suggests using a PSO as a model. Perhaps, for example, a preference relationship with anomalies that disobey the rules set by Luce. Luce contended that a function p : V ---+ R together with a constant 6 > 0 models a preference relation on V by, for x, y E V, xis preferred over y if f(x) > f(y) + 6. So, one could interpret f as some sort of ranking function and x is preferred over y if x is ranked sufficiently higher than y; thus, there will be no preferrence between members of V that are similar. If, however, 133

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there were a subset of V with the property that preference between elements did not obey the structure imposed by the choice of o, or whose prefernce structure obeyed a much smaller constant, this may be such a situation conducive to PSO model. In particular, we note that a PSO is the interplay of a linear order on g-1(N) and a semiorder on g-1(P), but not the intersection of two such orders. 134

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7. Concluding Remarks and Future Directions In this final chapter, we discuss some open problems and suggest programs for future study. In Chapter 5 we saw that if given a unit interval bigraph, there are many ways to represent it. Part of the focus of this chapter is to illustrate other ways to represent some of the different classes of graphs we have discussed. Results like Theorem 5.23 and Corollary 6.9 show that sometimes a different representation can be conducive to easier proofs. Indeed, were it not for Theorem 3.6, in particular the relationship between interval bigraphs and circular arc graphs, what is known about the structure of interval bigraphs may still be further from the truth. We will allude to some open problems in an order like the presentation of the material in this thesis. 7.1 Interval k-graphs and Probe Interval Graphs Other than Theorem 2.6 there are no characterizations for interval k-graphs in general. Theorem 3.3 gives a complete characterization for the interval kgraphs that are cycle-free via forbidden subgraphs, but the discussion in Chapter 3 regarding the state of the forbidden subgraph characterization for interval bigraphs suggests that, even for the interval 3-graphs, this type of characteri zation will be very difficult. One way to proceed here would be to identify the largest class of interval k-graphs that are cocomparability graphs and explore the corresponding order given by their complements. This tactic is a good one because it narrows the focus, and will bring something moreto the problem; in 135

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this case, the theory of partially ordered sets. To these ends, we define the next class of interval k-graphs. Let G be an interval k-graph with I the collection of intervals that represent it and Ii C I, for 1 i k, the set consisting of intervals that correspond to all vertices with color i; that is, Ii is interval class i. G is weakly proper if in each interval class no interval properly contains, nor is properly contained in, another from that class. We believe that the class of weakly proper interval k-graphs is the largest class of interval k-graphs that are cocomparability graphs. Theorem 7.1 If G is a weakly proper interval k-graph, then it is a cocompara:. bility graphs. Proof: Let G be a weakly proper interval k-graph and assume that all in tervals have distinct left endpoints. Observe that the construction of a function graph from a proper interval k-graph in Theorem 5.23 will create a function rep resentation for a weakly proper interval k-graph as well. Hence weakly proper IkGs are cocomparability graphs since function graphs are, see [18]. As for probe interval graphs, there is also only one class that is completely characterized by forbidden subgraphs, the cycle-free graphs, see [43]. Theorem 7.2 (Sheng, 1999) A cycle-free graph is a probe interval graph if and only if it has no subgraph isomorphic to H10 of Figure 3.1 or NL10 of Figure 3.3. For reasons similar to those for defining weakly proper interval k-graphs, we define a weakly proper probe interval graph to be a probe interval graph with 136

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no nonprobe interval properly containing, nor properly contained in, another nonprobe interval. We believe the class of weakly proper probe interval graphs is the largest class of probe interval graphs that are cocomparability graphs. Theorem 7.3 If G is a weakly proper probe interval graph, then it is a cocomparability graph. Proof: Let G = (P, N, E) be a weakly proper probe interval graph with probe interval representation where Lp is the collection of intervals representing probes and IN the collection of intervals representing nonprobes each belonging toR Assume with no loss of generality that the left (and hence right) endpoints of intervals in IN are distinct. We will define a parallelogram representation for G. Let 1 and 2 be a two parallel copies of R situated in JR2 For each nonproben define a line segment Ln with one end corresponding to l(n) on and the other to r(n) on 2 Since G is weakly proper, no two Ln's will intersect. For each probe p E P define a rectangle Rp with corners l(p), r(p) on 1 and l(p), r(p) on 2 Clearly, two Rp's intersect if and only if the corresponding in tervals intersect if and only if the corresponding vertices are adjacent in G. Also Rp and Ln intersect if and only if I(p) ni(n) =1-0 if and only if pn E E. Thus, the construction gives a parallelogram representation for G, and since parallelogram graphs are cocomparability graphs, by [29], G is also a cocomparability graph. Figure 7.1 illustrates the construction in Theorem 7.3, with thick black intervals representing probe intervals and the vertical lines are intended to clarify the construction. 137

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Figure 7.1: A parallelogram representation for a weakly proper PIG. Neither the benefits, if any, of the function representation for interval kgraphs, nor the parallelogram representation for probe interval graphs have been explored. The proof of Theorem 2.8 is very similar to the one for interval graphs that typically is given in text books containing introductory material on intervCI,l graphs. This suggests that there may be algorithms for finding parameters like clique covering number, independent set number, and etc. for interval k-graphs that parallel in "niceness" those for interval graphs; see [20], for example. 7.2 Interval Bigraphs We have discussed the lack of a forbidden subgraph characterization for interval bigraphs in Chapter 3. Indeed, as this is written, there is no working conjecture, for such a characterization. The author believes that there is a nice proof for Theorem 3.6 that ties the results together in a cycle of implications. Such a proof may be useful if new directions or techniques are used to obtain the proof. In particular, one might be able to discern what properties prohibit a 2-clique circular arc graph from having a representation with no two arcs covering 138

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the host circle, by way of proving (b) ==> (c), or (b) ==> (d) of Theorem 3.6. The tactic typically employed, not only by the author, but by any graph theorist seeking to characterize some class of graphs, is to narrow the focus suf ficiently to make headway. The characterization for cycle-free interval bigraphs serves as an example of the practicality of this approach. Here is how we pro pose to limit focus in another way. A tournament is a directed complete graph; that is, for every pair of vertices x, y either x -+ y or y -+ x. Recall that an interval digraph is a directed graph with an ordered pair of intervals (Su, Tu) corresponding to each vertex u such that u -+ v if and only if Su n Tv # 0. We propose that one examine which tournaments have an interval digraph repre sentation. A tournament T with adjacency matrix Jvf has an interval digraph representation if and only if the bipartite graph obtained from M is an interval bigraph. This follows from the equivalence of the models discussed Chapter 1. This restriction of perspective may be fruitful, not only in that it is a difficult and interesting problem in and of itself, but also because it may lead to new forbidden subgraphs. A graph on n vertices is cycle extendable if for every cycle C on less than n vertices there is another cycle C* that contains every vertex of C together with one additional vertex. A graph is Hamiltonian if it has a cycle that contains all vertices as a subgraph. It has been shown in [8] that Hamiltonian interval graphs are cycle extendable. This supports the conjecture of Hendry in [25] that Hamiltonian chordal graphs are cycle extendable. In [26] the bipartite analogue of cycle extendability is defined: a bipartite graph on n vertices is bi-cycle extendable if for every cycle C on fewer than n vertices, there is another cycle C* 139

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that contains all vertices of C together with two additional vertices. Since inter val k-graphs are weakly chordal, interval bigraphs are chordal bipartite, that is, they contain no induced cycle of length 6 or greater. Thus, for reasons similar to those that ultimately show that Hamiltonian interval graphs are cycle extend able, it seems reasonable to conjecture that Hamiltonian interval bigraphs are bi-cycle extendable. Since interval k-graphs are not chordal, the same suggestion can not be made about Hamiltonian interval k-graphs, for k > 2. 7.3 Interval Point Bigraphs Settling Conjecture 4.5 to a certain extent relies on no surprises coming from the examination of interval bigraphs. For example, if some unforeseen substructure that is forbidden as an interval bigraph is discovered, this may elicit a reworking of the conjectured list. For now, the conjecture stands as. such because of the tedious, case by case arguments that seem to be the only way to proceed. 7.4 Unit Interval k-graphs Unit interval k-graphs are characterization free for k > 2. They are cocomparability graphs since weakly proper interval k-graphs are, but not .conversely. This observation points one away from the cocomparability graph perspective, since, apparently, only in the 2-chromatic case are they precisely cocomparability graphs. We remind the reader of the discussion pertaining to Figure 5.1. In particu lar, it is clear that a certain algorithm for interval class assignment, a modified coloring algorithm, will be needed to deal with unit interval k-graphs. Also, we remind the reader that Figure 5.5 comes nowhere near a complete list of forbidden subgraphs. 140

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7.5 Unit Probe Interval Graphs Settling Conjecture 6.20 is a matter of time. Beyond that we propose to examine which unit probe interval graphs are 2-trees. A 2-tree is defined induc tively as follows: A K2 is a 2-tree. If G is any 2-tree witheE E(G) and if H rv K3 is vertex disjoint from G with e' E E(H), then the graph formed from G and H by identifying edges e and e' (along with their endpoints) is also a 2-tree. The 2-trees Hand Gin Figure 7.2 are not unit probe interval graphs. Notice that H has an asteroidal triple, hence by Theorem 6.2, is not a unit probe interval graph. We will show that G is not a unit probe interval graph using Lemma 6.10 and Lemma 6.11. We proceed by way of contradiction. Assume G = (P, N, E) is a unit probe interval graph. If x E N, then a, b, c, d, e, f E P, and in any representation I (c) U I (d) I ( x), contradicting that all intervals are unit length. So, x E P and to avoid an induced G2 of Figure 6.5, there must be at least one non probe among the vertices a, b, c, d, e, f. If {a, c, e, x} C P, then G-{b, d, f} is isomorphic to G2 of Figure 6.5; if {b, d, J, x} C P, then G -{a, c, e} rv G2 giving a contradiction either way. By definition of a probe interval graph, at least one of each pair {a, b}, {c, d}, {e, f} must be a probe. We now have the possibilities (a.), (b.), and (c.) in Figure 7.3 for subgraphs induced on the probes of G. Case (a.) gives a G2 In case (b.), deleting vertex y gives a G2 In case (c.), deleting vertices y and z gives a G2 Since a unit probe interval graph cannot have an induced G2 and we have covered all possible P, N 141

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assignments, G is not a unit probe interval graph. X c d H G Figure 7.2: 2-trees that are not UPIGs. (a.) (b.) (c.) y y Figure 7.3: Possible subgraphs induced on probes of G of Figure 7.2 7.6 A Final Remark On Probe Interval Graphs The author contends that there is no hope for a reasonable forbidden subgraph list for probe interval graphs. To illustrate this, we will restrict our attention to asteroidal triple-free graphs, and use only Proposition 2.1 to concoct an infinite class of graphs that are not probe interval graphs. Refer to Figure 7.4; the black vertices must be probes. In the top graph, observe that the struc-ture on the ends allows only one way to partition its vertices into probes and nonprobes in accord with Proposition 2.1, and this is the way that is shown. Then, working from left to right, we see that the partition given is forced and 142

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that we have two adjacent nonprobes at the right-hand end of the graph. It is easy to check that the removal of any vertex will allow for a reassignment that is conducive to a probe interval representation. For the other graphs, they are constructed in a way similar to the top graph and so that no asteroidal triple is formed. Each graph has a dominating path, that is, a path such that each vertex is either on it or adjacent to a vertex on it, and so, by [9], they are asteroidal triple-free. Figure 7.4: Asteroidal triple-free, 3-chromatic, minimal, forbidden PIGs. The principle for constructing these can be used to create infinitely many minimal forbidden PIGs. 143

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