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On characterizations and structure of interval digraphs and unit probe interval graphs

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On characterizations and structure of interval digraphs and unit probe interval graphs
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Dasgupta, Shilpa
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Denver, CO
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University of Colorado Denver
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English

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Doctorate ( Doctor of philosophy)
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University of Colorado Denver
Degree Divisions:
Department of Mathematical and Statistical Sciences, CU Denver
Degree Disciplines:
Applied mathematics
Committee Chair:
Lundgren, J. Richard
Committee Members:
Jacobson, Michael S.
Ferrara, Michael
Cherowitzo, William E.
Brown, David E.

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University of Colorado Denver
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Auraria Library
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Copyright Shilpa Dasgupta. Permission granted to University of Colorado Denver to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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ON CHARACTERIZATIONS AND STRUCTURE OF INTERVAL DIGRAPHS
AND UNIT PROBE INTERVAL GRAPHS
by
Shilpa Dasgupta
M.S., University of Colorado Denver, 2007
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics
2012


This thesis for the Doctor of Philosophy degree by Shilpa Dasgupta has been approved by
J. Richard Lundgren, Advisor and Chair Michael S. Jacobson Michael Ferrara William E. Cherowitzo David E Brown
Date
n


Dasgupta, Shilpa (Ph.D., Applied Mathematics)
On Characterizations and Structure of Interval Digraphs and Unit Probe Interval Graphs
Thesis directed by Professor J. Richard Lundgren
ABSTRACT
Interval graphs and their variations have been studied extensively for the last 50 years from both a theoretical standpoint and due to their importance in applications. In this thesis we will explore several variations of interval graphs and unit interval graphs including interval digraphs, interval bigraphs and probe interval graphs. Interval graphs were mathematically introduced by Hajos [22], The origin of interval graphs may also be regarded as an application to the research of Benzer in 1959 [1] in his study of the structure of bacterial genes. Many nice characterizations have been found for interval graphs, but the same cannot be said about the various generalizations. The work presented in this thesis will indicate that characterization of the broadest classes of the generalizations poses a very difficult problem and so the focus in this thesis is on subclasses of interval digraphs, interval bigraphs, and probe interval graphs.
Interval digraphs were introduced by Sen et al. in [39] (1989). In 1998 West gave adjacency matrix characterizations of interval digraphs and unit interval digraphs
[42], So far, in the most generic sense, no forbidden subgraph characterization of interval digraphs has been been found, but those tournaments which are interval digraphs have been characterized in various ways by Brown, Busch and Lundgren in 2007. In this thesis we generalize some of their results to other classes of interval digraphs.
A natural extension of interval graphs, called interval bigraphs, were introduced by Harary, Kabell, and McMorris [23] in 1982. Unit and proper interval graphs were
iii


introduced and characterized by Roberts in 1969 [36]. He proved that the classes of proper interval graphs and unit interval graphs coincide. Proskurowski and Telle [33] extended the idea of proper interval graphs to p-proper interval graphs. Recently Beyerl and Jamison introduced p-improper interval graphs. They focussed on a special case of p-improper interval graphs. Here we generalize and obtain similar results for p-improper interval bigraphs.
The probe interval graph was invented in order to aid with the task called physical mapping faced in connection with the human genome project, cf. work of Zhang and Zhang et al. [43], [45], [44], No generalized characterization of probe interval graphs have been found so far. Li Sheng first characterized probe interval graphs for trees[41], Following this characterization of cycle-free probe interval graphs, in 2009 Brown, Sheng and Lundgren gave a characterization for cycle-free unit probe interval graphs [15]. Recently Brown and Langley characterized unit probe interval graphs for bipartite graphs [10]. In this thesis we give a characterization of 2-trees that are unit probe interval graphs. In 2005 Przulj and Cornell found at least 62 distinct minimal forbidden induced subgraphs for probe interval graphs that are 2-trees [34], More recently Brown, Flesch and Lundgren extended the list to 69 and also gave a characterization [7]. In this thesis we follow the idea of Przulj and Cornell for the next best case which is for 2-trees. The restriction to 2-trees is a reasonable thing to do because of the progression of the Sheng, Brown-Sheng-Lundgren, and Brown-Langely results, and also because of the Corneil-Przulj and Brown-Flesch-Lundgren results. We first characterize 2-caterpillars and interior 2-caterpillars in terms of forbidden induced subgraphs and show that 2-trees that are unit probe interval graphs have to be interior 2-caterpillars. Then we look at 2-paths that are unit probe interval graphs and characterize them. Using similar ideas we subsequently characterize interior 2-caterpillar which are unit probe interval graphs. Finally we use these results to get a complete characterization of 2-trees which are unit probe interval graphs.
IV


The form and content of this abstract are approved. I recommend its publication.
Approved: J. Richard Lundgren
v


DEDICATION
Mom and dad and my sweethearts Isha, Sheila, Anoushka and Bubai.


ACKNOWLEDGMENTS
First of all I would like to thank my adviser, Dr. J. Richard Lundgren, for guiding, mentoring and engaging me in mathematics. I would also like to thank my professors, friends and colleagues with whom I’ve worked over the past few years. Lastly, and certainly most importantly, I thank my family, especially my sister for her love and support.
Vll


TABLE OF CONTENTS
Figures .................................................................... ix
1. Introduction................................................................ 1
1.1 Background................................................................ 1
1.2 Introduction to the research.............................................. 5
2. Interval Digraphs........................................................... 7
2.1 Introduction.............................................................. 7
2.2 Motivation from Interval Tournaments..................................... 11
2.3 Digraphs with a Transitive (n — 2)- subtournament........................ 14
2.4 More than one nontrivial strong component ............................... 22
3. Interval Bigraph Impropriety.............................................. 27
3.1 Introduction............................................................. 27
3.2 Impropriety and Weight of Interval Bigraphs.............................. 29
3.3 p-critical interval bigraphs............................................. 36
3.4 Conclusions and future work.............................................. 47
4. Characterization of unit probe interval 2-trees.......................... 48
4.1 Introduction............................................................. 48
4.2 Preliminaries............................................................ 49
4.3 Characterization of 2-caterpillars....................................... 56
4.4 Forbidden subgraphs for 2-paths which are unit probe interval graphs . . 61
4.5 2-path unit probe interval graph characterization........................ 70
4.6 forbidden subgraphs for interior 2-caterpillars which are unit probe interval
graphs ............................................................ 119
4.7 Characterization of interior 2-caterpillars which are unit probe interval
graphs ............................................................ 130
References.................................................................. 172
viii


FIGURES
Figure
2.1 Interval digraph............................................................... 9
2.2 Zero partition................................................................ 10
2.3 Interval bigraph.............................................................. 10
2.4 Insect........................................................................ 11
2.5 Bigraph representation of a digraph........................................... 11
2.6 D-v........................................................................... 13
2.7 ATE in the bigraph............................................................ 15
2.8 6-cycle....................................................................... 16
2.9 10-cycle...................................................................... 16
2.10 Induced 6-cycle in the bigraph representation................................ 17
2.11 More restrictions on the arcs ....................................... 17
2.12 Example of a digraph satisfying a condition of the theorem 2.3.1 and its
zero partition .............................................................. 25
2.13 Example of a digraph satisfying a condition of the theorem 2.3.2 and its
zero partition............................................................... 26
2.14 Example of a digraph that fails the hypothesis of theorem 2.3.2 since v
beats v” and hence forms a 6-cycle in its bigraph representation......... 26
2.15 Strong components............................................................ 26
3.1 K1>3.......................................................................... 29
3.2 p-Improper Interval Graph .................................................... 30
3.3 0-Improper Interval Bigraph ....................................... 30
3.4 Forbidden subgraphs for proper Interval bigraph with their interval representations to force containment from X .................................. 31
3.5 Interval representations of Bl, B2, B3 to force containment from Y . . . 31
3.6 Impropriety of imp(B(f>1) = 2, imp(B(f>2) = 1................................. 33
IX


3.7 Exterior local components .................................................. 33
3.8 Interval bigraph with impropriety 1 34
3.9 Interval bigraph with impropriety 2 34
3.10 Calculation of weight ..................................................... 35
3.11 Interval bigraph with weight 2 35
3.12 Changes in weight due to variation in definition .......................... 37
3.13 Interval bigraph where wt(X) = 0 < irrip(B) = 1 37
3.14 Interval bigraph where wt(X) = 1 = irrip(B) = 1 38
3.15 Illustrations of balance and p-criticality ................................ 39
3.16 Choice..................................................................... 40
3.17 Possible structures of an interval bigraph with a balanced and p-critical
partite set................................................................ 44
4.1 Examples of some 2-paths.................................................... 50
4.2 On the left is a 2-caterpillar, and on the right is an interior 2-caterpillar 51
4.3 4-fan....................................................................... 53
4.4 Unit probe interval representation of a 4-fan............................... 53
4.5 An illustration where unit probe interval representation of a 4-fan fails to
work....................................................................... 53
4.6 FI.......................................................................... 54
4.7 Probe interval representation of a 3-sun or FI.............................. 55
4.8 El.......................................................................... 56
4.9 Forbidden subgraphs for 2-caterpillar ...................................... 58
4.10 Construction 1............................................................ 58
4.11 Construction 2............................................................ 59
4.12 Construction 3............................................................ 59
4.13 Six 2-trees called Ai s ................................................... 60
4.14 F2...................................................................... 63
x


4.15 F3......................................................................... 63
4.16 F4......................................................................... 64
4.17 F5......................................................................... 65
4.18 F6......................................................................... 66
4.19 F7......................................................................... 67
4.20 F8......................................................................... 67
4.21 F9......................................................................... 68
4.22 F10 ....................................................................... 69
4.23 Fll ....................................................................... 70
4.24 Non-isomorphic 2-paths of length 6......................................... 72
4.25 Construction that shows possible merge of two-4-fans....................... 73
4.26 Construction of two edge-consecutive 4-fans................................ 76
4.27 All possible two edge-consecutive 4-fans and their representations .... 76
4.28 Formation of three edge-consecutive-4-fans................................. 77
4.29 Formation of three edge-consecutive-4-fans................................. 77
4.30 Formation of three edge-consecutive-4-fans................................. 78
4.31 three edge-consecutive-4-fans ............................................. 78
4.32 Formation of two vertex-consecutive-4-fans................................. 80
4.33 The only structure possible for two vertex-consecutive-4-fans.............. 80
4.34 Representations of three vertex-consecutive-4-fans......................... 81
4.35 Another representation of three vertex-consecutive-4-fans.................. 81
4.36 Three possible structures of two edge-consecutive 4-fans .................. 82
4.37 Straight 2-path............................................................ 83
4.38 Staircase.................................................................. 84
4.39 2-snake ................................................................... 84
4.40 Extended-staircase+staircase .............................................. 85
4.41 Staircase representation................................................... 85
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4.42 2-snake representation.................................................... 86
4.43 extended-staircase+staircase representation............................... 86
4.44 Four edge-consecutive 4-fans, structure 1................................. 91
4.45 Four edge-consecutive 4-fans, structure 2................................. 92
4.46 Four edge-consecutive 4-fans, structure 3................................. 92
4.47 Example of merges for different values of n .............................. 97
4.48 n = 1 : edge-consecutive merge for 2-paths.............................. 97
4.49 n = 1 : edge-consecutive merge for 2-paths.............................. 98
4.50 n = 1 : edge-consecutive merge for 2-paths.............................. 99
4.51 n = 2 : edge-consecutive merge for 2-paths.............................. 100
4.52 n = 2 : edge-consecutive merge for 2-paths.............................. 101
4.53 n = 2 : edge-consecutive merge for 2-paths.............................. 101
4.54 n = 2 : edge-consecutive merge for 2-paths.............................. 102
4.55 n = 2 : edge-consecutive merge for 2-paths.............................. 103
4.56 n = 3 : edge-consecutive merge for 2-paths ............................. 103
4.57 n = 3 : edge-consecutive merge for 2-paths ............................. 104
4.58 n = 3 : edge-consecutive merge for 2-paths.............................. 104
4.59 n = 0 : vertex-consecutive merge for 2-paths............................ 108
4.60 n = 1 : vertex-consecutive merge for 2-paths............................ 109
4.61 n = 2 : vertex-consecutive merge for 2-paths............................ 110
4.62 n = 3 : vertex-consecutive merge for 2-paths ..................... 110
4.63 vertex-consecutive-edge-consecutive for 2-path........................... 112
4.64 vertex-consecutive-edge-consecutive for 2-path........................... 113
4.65 n = 1 : vertex-consecutive-edge-consecutive for 2-path.................. 113
4.66 n = 2 : vertex-consecutive-edge-consecutive for 2-path.................. 114
4.67 n = 2 : vertex-consecutive (2-snake)-edge-consecutive for 2-path........ 114
4.68 Straight-2-path+end-2-leaves ............................................ 116
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4.69 Staircase+end-2-leaves ................................................... 117
4.70 2-snake+end-2-leaves...................................................... 118
4.71 Non-probe restriction in 4-fans .......................................... 120
4.72 Non-probe restriction .................................................... 121
4.73 Unit probe interval representation ....................................... 122
4.74 3-fan with a 2-leaf....................................................... 122
4.75 E2........................................................................ 123
4.76 probe interval representation for E2...................................... 123
4.77 E3........................................................................ 124
4.78 E4........................................................................ 125
4.79 E5........................................................................ 126
4.80 Probe interval representation of E5....................................... 126
4.81 E6........................................................................ 127
4.82 Probe interval representation of E6....................................... 127
4.83 E7........................................................................ 128
4.84 E8........................................................................ 129
4.85 E9........................................................................ 129
4.86 E10 ...................................................................... 130
4.87 Group-Ell- ul,u2 are end 2-leaves of £>* and Bj ...................... 131
4.88 Possible places of 2-leaves............................................... 132
4.89 Representation of 2-snake with all possible 2-leaves...................... 133
4.90 Representation of extended staircase with all 2-leaves ................... 133
4.91 Representation of staircase with all 2-leaves............................. 134
4.92 Representation of vertex-consecutive 4-fans with all 2-leaves............. 134
4.93 Representation of 2-snake with all 2-leaves............................... 135
4.94 Representation of extended-staircase and staircase with all 2-leaves ... 135
4.95 Edge-consecutive 4-fans with all 2-leaves................................. 136
xiii


4.96 4-fan with all 2-leaves .............................................. 136
4.97 Vertex consecutive 4-fans with all 2-leaves........................... 142
4.98 n = 1 : h............................................................. 150
4.99 n = 1 : h............................................................. 151
4.100n = 2: h............................................................. 152
4.101 n = 3 : h............................................................. 153
4.102n = 4 : /i............................................................. 154
4.103 Representation of straight 2-path with all 2-leaves ................. 154
4.104n = 0 : I2............................................................. 159
4.105n =1: I2............................................................. 160
4.106n = 2: I2............................................................. 160
4.107n = 3 : I2............................................................. 161
4.108n = 0 : h............................................................. 165
4.109n = 1 : h............................................................. 166
4.110ra = 2: h............................................................. 166
4.111 Interior 2-caterpillars with fewer 2-leaves .......................... 167
xiv


1. Introduction
1.1 Background
Interval graphs and their variations have been studied extensively for the last 50 years from a theoretical standpoint and due to their importance in applications. In this thesis we will explore several variations of interval graphs namely interval digraphs, interval bigraphs and probe interval graphs. The work on interval bigraphs will focus on p-improper interval bigraph and the work on probe interval graphs will be on unit probe interval graphs.
Let a graph G have vertex set V(G) and edge set E(G). If x,y E V(G) are adjacent, then we denote xy E E(G). If G is bipartite, we denote the partitions of the vertex set as V(G) = {X Ub}. A graph is interval if to every vertex, v E V(G), we can assign an interval of the real line, Iv, such that xy E E(G) if and only if Ix H Iy ^ 0. Interval graphs were theoretically introduced in 1957 by Hajos [22] and also appeared in applied research by Benzer in 1959 [1] in his study of the structure of bacterial genes. Interval graphs were characterized by the absence of induced cycles larger than 3 and asteriodal triples by Lekkerkerker and Boland [26] in 1962. An asteroidal triple (AT) in G is a set A of three vertices such that between any two vertices in A there is a path in G that avoids all neighbors of the third.
Other useful characterizations of interval graphs were given by Gilmore and Hoffman in 1964 [19] and Fulkerson and Gross [17]. Extensive study and research has been done on interval graphs for several decades by both mathematicians and computer scientists. These graphs are used to provide numerous models in diverse areas such as genetics, psychology, sociology, archaeology, or scheduling. For more details on interval graphs and their applications, see books by Roberts [36], Golumbic [20] and Mckee and McMorris [29].
Interval digraphs were introduced by Sen et al. in [39](1989). Langley, Lundgren,
Merz gave results on Competition Graphs of Interval Digraphs (1995) in [25]. They
1


showed that the competition graph of an interval digraph is an interval graph and that every interval graph is in fact the competition graph of some interval digraph. Lin, Sen and West gave some interesting results on Interval digraphs and (0,l)-matrices in [27]. In 1998 West gave adjacency matrix characterizations of interval digraphs and unit interval digraphs [42], At this point there is no forbidden subgraph characterization of interval digraphs, but recently, in 2007, interval tournaments were characterized by a complete list of forbidden subtournaments by Brown, Busch, and Lundgren [6]. In chapter 2 we generalize some of the results from this paper.
A natural extension of interval graphs, called interval bigraphs, were introduced by Harary, Kabell, and McMorris [23] in 1982. Let G be a bipartite graph with bipartition X U Y; we may write G = (X, Y, E) to denote this. A bipartite graph G is an interval bigraph if to every vertex, v G V(G), we can assign an interval of the real line, Iv, such that xy G E(G) if and only if Ix C\Iy yt 0 and x G X and y eY. Interval bigraphs have been studied by several authors ([9], [14], [23], [24], [27], and [31]). Initially it was thought that the natural extension of asteriodal triples of vertices to asteriodal triples of edges along with induced cycles larger than 4 would give a forbidden subgraph characterization [23]. However, Muller [31] found insects and Hell and Huang [24] found edge asteriods and bugs as forbidden subgraphs, and to date a complete characterization remains elusive. Three edges a, c and e of a graph G form an asteriodal triple of edges (ATE) if for any two there is a path from the vertex set of one to the vertex set of the other that avoids the neighborhood of the third edge. Cycle free interval bigraphs were characterized by Brown et al in 2001 [14], and ATEs played a significant role in that characterization. In 2002 a generalization of interval bigraph called an interval fc-graph was introduced by Brown et al [8]. More recently, in 2010, Lundgren and Tonnsen characterized 2-trees that are interval fc-graphs [28].
2


A graph is a probe interval graph if there is a partition of V(G) into sets P and N and a collection {Iv : v G V(G)} of intervals of R such that, for u,v G V(G), uv G E(G) if and only if Iu n Iv ^ (f) and at least one of u or v belongs to P. The sets P and N are called the probes and nonprobes, respectively. An interval graph is a probe interval graph with an empty non-probe set and this class of graphs has been studied extensively. The probe interval graph model was invented in order to aid with the task called physical mapping faced in connection with the human genome project (cf. work of Zhang and Zhang et al. [43], [45], [44]). The paper by Mc-Morris, Wang and Zhang [30] has results similar to those for interval graphs found in [17] by Fulkerson and Gross and [20] by Golumbic. For example interval graphs are chordal while probe interval graphs are weakly chordal and maximal cliques are consecutively orderable in interval graphs while quasi-maximal cliques works in probe interval graphs. In 1999 Li Sheng first characterized cycle-free probe interval graphs [41] The classes of graphs related to probe interval graphs are discussed in [11] by Brown and Lundgren, [8] by Brown, Flink and Lundgren, and [21] by Golumbic and Lipshteyn. Relationships between bipartite probe interval graphs, interval bigraphs and the complements of circular arc graphs are presented in [11], The problem of characterizing generic probe interval graphs in terms of forbidden subgraphs for now appears to be difficult. Since trees were the only class of graphs where probe interval graphs were characterized, a natural next step was to look at a class of 2-trees. In 2005 Przulj and Corneil attempted a forbidden subgraph characterization of 2-trees that are probe interval graphs and they found at least 62 distinct minimal forbidden induced subgraphs for probe interval graphs that are 2-trees [34], More recently Brown, Flesch and Lundgren extended the list to 69 and gave a characterization in terms of sparse spiny interior 2-lobsters [7].
In the last 40 years a subclass of interval graphs has been investigated and studied
3


extensively. This class is the class of unit interval graphs introduced by Roberts in 1969. A unit interval graph is an interval graph with all intervals in some interval representation having the same length. A proper interval graph is an interval graph for which there is an interval representation with no interval properly containing another. Roberts [36] proved that the classes of proper interval graphs and unit interval graphs coincide and he showed that interval graphs that contain no Jd1;3 are unit interval graphs. In 1999 Bogart and West gave a shorter proof of the same result of the equality of proper and unit for interval graphs [4], They gave a constructive proof of this result, where proper intervals are gradually converted into unit intervals. Much more recently in 2007 Gardi gave a much shorter new constructive proof of Roberts original characterization of unit interval graphs [18].
A unit interval bigraph is an interval bigraph with all intervals in some interval representation having the same length and a proper interval bigraph is an interval bigraph for which there is an interval representation with no interval containing another properly. Several characterizations of proper interval bigraphs have been found in the last decade including one by Lin and West [27]. The idea of proper interval graphs was naturally extended to p-proper interval graphs by Proskurowski and Telle [33]. The p-proper interval graphs are graphs which have an interval representation where no interval is properly contained in more than p other intervals. Any 0-improper interval graph is a proper interval graph and it is easy to check that X13 is a is a 1-improper interval graph. In chapter 3 we will generalize the class of p-improper interval graph to p-improper interval bigraph. Unit interval digraphs were characterized in 1997 by Lin et al [27]. In 2002, Brown et al [9] conjectured a characterization of unit interval bigraphs. This conjecture was proved by Hell and Huang [24] in 2004. More recently in 2011 Brown and Lundgren gave several additional characterization of unit interval bigraphs [12], Currently Brown, Flesch and Lundgren are working on unit interval fc-graphs for 2-trees. Following up Sheng’s characterization of cycle-free
4


probe interval graphs, in 2009, Brown, Lundgren and Sheng gave a characterization of cycle-free unit probe interval graphs [15]. Two natural extensions arise, the first is to characterize bipartite graphs that are unit probe interval graphs and secondly to characterize 2-trees that are unit probe interval graphs. Recently Brown and Langley characterized unit probe interval graphs for bipartite graphs [10]. In chapter 4 we give a characterization of 2-trees that are unit probe interval graphs.
1.2 Introduction to the research
We will introduce notation as we move through the thesis. At the initial stage we will use conventional definitions. As mentioned earlier the study of interval graphs has long been a much researched area in Graph Theory. The majority of work in this thesis is an effort to characterize various classes of variations of interval graphs and unit interval graphs. In chapter 2 we study different classes of digraphs and determine the scenarios when they are bound to be interval. Since the problem of finding a complete characterization of interval digraphs is extremely difficult, we concentrate on specific classes of interval digraphs. We focus our research in this chapter on a paper of Brown, Busch, and Lundgren [6] where a complete characterization of interval tournaments is found. We look at oriented graphs with certain properties and fold restrictions that need to be imposed on its structure for it to be interval. We fold several classes of directed graphs on n vertices which are interval with restrictions such as containing transitive sub-tournaments on (n — 2) or (n — 1) vertices and specific adjacencies between vertices.
As stated earlier, another natural extension of interval graphs, called interval bigraphs, were introduced by Harary, Kabell, and McMorris in 1982. The p-improper interval graphs, where no interval contains more than p other intervals, were investigated by Beyerl and Jamison [3]. In chapter 3 we extend the idea by introducing p-improper interval bigraphs, where no interval contains more than p other intervals of the same partite set. In this chapter we determine restrictions on the structure of
5


an interval bigraph for it to be a p-improper interval bigraph. We also study special classes of p-improper interval bigraphs.
In chapter 4, we look at unit probe interval graphs. Since they have been characterized for trees and bipartite graphs , we follow the techniques of Przulj and Corneil for the next best case which is for 2-trees. We characterize 2-trees that have a unit probe interval representation. In order to accomplish this we first characterize 2-caterpillars and interior 2-caterpillars in terms of forbidden induced subgraphs and show that 2-trees that are unit probe interval graphs have to be interior 2-caterpillars. At this point we realized that the problem was much more difficult than we originally anticipated it to be. So we look at 2-paths that are unit probe interval graphs and characterize them. Then we use this characterization to fold the characterization of interior 2-caterpillar which are unit probe interval graphs. Finally we use these results to get a complete characterization of 2-trees which are unit probe interval graphs.
6


2. Interval Digraphs
2.1 Introduction
A directed graph, or digraph is a graph in which each edge is assigned a direction. An arc (directed edge) from vertex u to vertex v will be denoted u w, and we will say that u beats v. The set of vertices of a digraph D will be denoted V(D), and its size will be denoted | V(D) |. A subdigraph of D is a digraph consisting of a subset of V(D), with only arcs from D between the vertices in this subset. Throughout this paper, we will only be considering digraphs that have no 2- cycles or loops; so for two vertices u,v E V(D), if u i—> v, then we cannot have v i—> u, or u = v
A tournament T is an oriented complete graph, so for any x,y E V(T) either x i y y or y i y x, but never both, and x e-> x can never happen. An interval tournament is a tournament that is an interval digraph. Naturally a subtournament of some digraph D is a subdigraph of D that happens to be a tournament.
A directed graph D is an interval digraph if for each vertex u there corresponds an ordered pair of intervals (SU,TU) such that for any two vertices u,v E V(D), u i—y Vj if and only if Su fl Tv yt

7


with a complete list of forbidden subtournaments by Brown, Busch, and Lundgren [6]. They show that a tournament on n vertices is an interval digraph if and only if it has a transitive (n — l)-subtournament. If a digraph D is not a tournament, then it may be an interval digraph even if it does not contain a transitive (n — l)-subtournament as a subdigraph, as long as there are specific restrictions on D.
We explore what restrictions we can place on D to guarantee that it is interval, and in particular we investigate a broader class of oriented graphs on n vertices that contain a transitive (n — 2)-tournament as a subdigraph. A complete characterization of interval digraphs appears to be really difficult but we managed to fold classes of oriented graphs that are interval digraphs. Thus we investigate different types of digraphs (most of which have characteristics in common with tournaments) to try and determine classes of digraphs that can be shown to have an interval representation.
For a graph G that is not a directed graph, an adjacency between vertices u, v E V (G) will be denoted astmti. A bipartite graph, or bigraph, B is a graph in which the vertices are partitioned into two sets X and Y, such that XU Y = V(B), and any two vertices u,v E V(B) can only be adjacent if one vertex from u, v is in X and the other is in Y (and this does not guarantee that they will be adjacent). A bipartite graph B is an interval bigraph if to each vertex there corresponds an interval such that two vertices are adjacent if and only if their corresponding intervals intersect and each of these two vertices belongs to a different partite set as shown in Figure 2.3. In 2004, Hell and Huang gave some interesting results on interval bigraphs [24], More interesting work has been done on interval bigraphs and other related graphs by Brown and Lundgren in 2006 [11]. Methods for recognition of interval bigraphs and interval digraphs have also been investigated in [31] by Muller. He gave a dynamic programming algorithm recognizing interval bigraphs (interval digraphs) in polynomial time. This algorithm recursively constructs a bipartite interval representation of a graph from bipartite interval representations of proper subgraphs. We will see
8


later that the models for interval digraphs and interval bigraphs are basically same. We use the equivalence of the models for interval digraphs and interval bigraphs in our investigation of which oriented graphs are interval digraphs.
An adjacency matrix of a digraph D, denoted A(D), is a 0, 1-matrix that has a 1 in the entry a^j (row i, column j) if and only if ig i—> Vjfor the two vertices Vi,Vj E V(D). A 0, 1-matrix has a zero partition if after independent row and column permutations every zero can be labeled as C or R, such that below every C is another C (except for C’s in the bottom row) and to the right of every R is another R (except for R’s in the far right column). Figure 2.2 gives a zero partition of a matrix.
We take into account important results on interval digraphs and interval bigraphs and point out a connection between the two to prove the significant results in this chapter.
The following three theorems are helpful tools needed to show certain digraphs are, or are not, interval.
d (Sd.Td) c .
Figure 2.1: Interval digraph
Theorem 2.1.1 (Sen, Das, Roy, West [39]). A digraph D is an interval digraph if and only if A(D) has a zero partition.
An asteroidal triple of edges, or ATE, is a set of three edges in a graph for which there is a path between any two of these edges that avoids the neighborhood of the third (the neighborhood of an edge is the set of vertices that are adjacent to one of its end-vertices).
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Figure 2.3: Interval bigraph
Theorem 2.1.2 (Muller [31]). If B is an interval bigraph, then the following hold:
(a) B has no induced cycle on more than 4 vertices;
(b) B has no asteroidal triple of edges in any induced subgraph;
(c) B has no insect 2-4 as an induced subgraph.
A digraph D can be represented as a bigraph B(D) by letting every vertex v E V(D) correspond to two vertices in B(D) (one from each partite set) xv E X and yv E Y, such that u i—> v in D if and only if xu EE yv in B, and this relation accounts for all the edges in B(D). The following theorem puts the concepts of interval digraphs and interval bigraphs together to help us identify digraphs that have no interval representation.
Theorem 2.1.3 If D is an interval digraph, then B(D) is an interval bigraph.
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INSECT
Figure 2.4: Insect
Digraph D
B(D)
Note: We get an induced 6-cycle and so D is not an Interval Digraph
Figure 2.5: Bigraph representation of a digraph
We can get this result by letting IXv = Sv and IVv = Tv for any v E V(D) where xv and yv are the corresponding vertices in B(D). So, essentially, the models for interval digraphs and interval bigraphs are the same.
2.2 Motivation from Interval Tournaments
Recently, interval tournaments have been characterized by Brown, Busch, Lund-gren [6] and we will use part of a main result proved by them (the following theorem) for our proofs.
Theorem 2.2.1 (Brown, Busch, Lundgren [6]). Suppose that T is a tournament on n 'vertices. Then the following are equivalent:
(a) T is an interval tournament;
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(b) T has a transitive (n — 1)-subtournament;
The natural question that rises from this characterization is ” What are the other types of digraphs that can be classified as interval?”. So we look at digraphs with qualities that are similar to interval tournaments. We know that an interval tournament on n vertices has a transitive (n — 1)-subtournament. So first we look at digraphs with this same characteristic and this leads us to a generalization of Theorem
2.2.1 (6) =>- (a) using a similar proof. The following theorem is a generalization of the previous theorem (b) =>- (a) and we use the concept of zero partition for its proof.
Theorem 2.2.2 If D is a digraph on n vertices with no loops or 2-cycles and a transitive (n — 1)-subtournament, then D is an interval digraph.
Proof:
Let D be a digraph on n vertices with no 2-cycles or loops, and suppose the induced subgraph D — v is a transitive tournament as shown in Figure 2.6. Label the vertices of D — v as vn~\, vn-2,..., V2, v\ such that if Vi i—> Vj, then i < j for
i-J e 1.2....// 1. In forming the adjacency matrix A(D — v), let the rows and
columns correspond to this ordering above (from top to bottom and left to right respectively).
Now A(D — v) is lower-triangular with 0’s on and above the diagonal and l’s below it. So in every row all the l’s are on the left and all the 0’s are to the right of them. Therefore we have a zero-partition for A(D — v) in which all the 0’s can be labeled R.
Now attach the v column on the far left and the v row on the bottom of A(D — v). Permute the rows of this new matrix A(D) so that all the l’s in the v column are at the top and all the 0’s are below them, which can now be labeled as C’s. Note that we do not need to move the v row (at the bottom) to do this since its first entry is
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0. Also note that this does not require us to change the R labeling of the 0’s from A(D — v) since each of these 0’s had a 0 to the right of it, and we have only permuted the rows.
We can now label the 0’s in the v row (on the bottom) as C’s, since none of them will have l’s below. All of the 0’s in A(D) have now been labeled as R or C, so we have a zero-partition, and Theorem 2.1.1 implies that D is an interval digraph.
D is an Interval Digraph
Figure 2.6: D — v
Note that if a digraph D on n vertices has a (n — l)-subdigraph that is transitive, it is not necessarily an interval digraph. We only know that D has an interval representation in this case if the transitive (n — 1)-subdigraph is actually a tournament. In fact, a transitive digraph itself may not be interval. An example of this is in Figure 2.8, which shows a transitive digraph D that is not interval (by Theorem 2.1.2) because its bigraph B(D) has an induced 6-cycle.
Suppose now for a digraph D, the induced subgraph D — v is the union of k disjoint transitive tournaments for some k E N'. We find that there do exist several such digraphs that are not interval digraphs. We found 6-cycles, 8-cycles, 10-cycles
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and even ATEs in the corresponding bigraphs of such digraphs.
In Figure 2.9, the induced 10-cycle in the bigraph B(D) shows that D is not an interval digraph. However, it has been showed that the 10-cycle is the largest induced cycle that can exist in the bigraph B(D) of such a digraph D [16]. Thus 6, 8, and 10-cycles are the only cycles in the bigraph B(D) that would produce forbidden structures. It is worth mentioning here that it has also been proved that the longest induced path in the bigraph B(T) of any transitive tournament T consists of three edges (and four vertices) [16].
Unfortunately, we also discovered that there are many ways that ATE’s can appear in the bigraphs corresponding to digraphs D in which D — v is a union of disjoint transitive tournaments. One example is shown below in Figure 2.7. We know that that if B(D) has an ATE, then B(D) is not an interval bigraph, and hence D is not an interval digraph. These results lead us to consider a different generalization, namely, digraphs D on n vertices such that the induced subdigraph on (n-2) vertices given by D-{v ,v"} is a transitive tournament.
2.3 Digraphs with a Transitive {n — 2)- subtournament
A digraph honn vertices with a transitive (n — 2)-subtournament can have many structures in its bigraph B(D) that prevent it from being an interval digraph. Figure 2.10 below is an example where we get a forbidden subgraph in B(D) (which is a 6-cycle) for which D-{v ,v"} is a transitive tournament. It is also worth-mentioning here that for this particular digraph D, adding any combination of the dashed arcs (or none of them) will result in the same induced 6-cycle in B(D). Altogether, there are a total of 36 possible combinations of these dashed edges that will result in this particular induced 6-cycle.
The example in the Figure 2.10 shows that we can easily end up with an induced
6- cycle in B(D) for a digraph D in which D-{v',v'} is a transitive tournament.
Furthermore this type of digraph D has many forbidden structures that will prevent
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v4
induced ATE
The darker edges xly2, x4y6, x6y8 are the 3 edges
Figure 2.7: ATE in the bigraph
it from having an interval representation (including many induced cycles of length at least 6 and many ATE’s in its bigraph). This suggests that perhaps we need more restrictions on D to find a type of digraph that we know is interval. The following theorem places more restrictions on the arcs directed between the vertices of the transitive (n — 2)-subtournament and the other two vertices v and v".
Theorem 2.3.1 Let D be a directed graph on n vertices with no 2-cycles or loops, and ■with a transitive (n — 2)-subtournament D* =D-{v , v'}. Suppose either (a) or (b) is true, but not both:
(a) Suppose no vertices in D* beat v or v , and v” beats a subset of the vertices in D* that v beats;
(b) Suppose no vertices in D* are beaten by v or v", and v" is beaten by a subset of the vertices in D* that beat v .
Then D is an interval digraph
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transitive digraph which is not interval
B(D)
D
B(D)
induced 6-cycle in the bigraph
Figure 2.8: 6-cycle
v5
Figure 2.9: 10-cycle
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Any combination of the broken arcs also forms a 6 cylcle in B(D) In total we can get 36 ways of forming an induced 6-cycle in B(D)
Figure 2.10: Induced 6-cycle in the bigraph representation
Figure 2.11: More restrictions on the arcs
Proof: Let D be a digraph on n vertices with no 2-cycles or loops, and suppose D*=D-{v , v"} is a transitive tournament. Thus the subdigraph D-v" has a transitive (/?. — 2)-subtournament, and by Theorem 2.2.1 it is an interval digraph. Theorem
2.1.1 now implies that A(D — v") has a zero partition. Assume that D* meets the conditions of either (a) or (b) above, but not both. The following cases must be considered:
(i) v , and v" have no arc between them,
(ii) i” has an arc directed to v ,
(iii) v has an arc directed to v".
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Arrange the rows and columns of A(D — v") as in the proof of Theorem 2.2.1 Now add the column and row corresponding to v at the far left (with column v to its right) and bottom of the matrix (with row v above), respectively. Recall from the proof of Theorem 2.2.1 that the submatrix A(D*) already has each of its Os labeled as R.
If (a) is true (but (b) is not):
(i) Let v , and v have no arc directed between them. Row v (just above row v") has its Os labeled as Cs, as in the proof of Theorem 2.2.1. Row v will have a 0 in every column entry that row v has a 0, and will also have Os in some column entries in which row v has Is (and Is in the other entries). So each 0 in row v" can be labeled as C, which will each be below another C or a 1. Also, columns v and v will have all 0 entries since no vertices in D* beat these two vertices, and each of these 0’s can be labeled as C. Now A(D) has a zero partition.
(ii) If v has a arc directed to v , then av»v< is the only entry that is different from
A(D) in (i) above, and it now becomes a 1. This 1 is a problem because it is below
the Os in column v , which had previously been labeled as Cs. So now move column v to the far right. Because no vertices from D* have directed arcs to v or v", the 1 at the bottom of column v is the only 1 in that column, and all the Os above it can be relabeled as Rs. Now A(D) has a zero partition.
(iii) If v has a directed arc to v”, then av'v» is the only entry that is different from
A(D) in (i) above, and it now becomes a 1. In the same manner as part (ii), move
column v to the far right and relabel all entries above entry av'v» (which are all Os) as Rs. There is also one 0 below av'v» in column v", which can be labeled as C. Now A(D) has a zero partition.
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If (b) is true (but (a) is not):
(i) Let v and v have no arc directed between them. The rows of A(D — v") have already been permuted (as in the proof of Theorem 2.2.1) so that the rows with a I in the column v (just right of column v") are at the top, with all the 0’s in this column below the l’s and labeled as C’s. Now permute these rows with l’s in column v so that the rows with l’s in column v" are at the top with 0’s underneath (note that column v will still have all its l’s above its 0’s). Now all the 0’s are below the l’s in column v and v , and each of these 0’s can be labeled as C. The rows v and v will be composed entirely of 0’s, which can be labeled as C’s, since no vertices in D* are beaten by these two vertices. Now A(D) has a zero partition.
(ii) If v has a directed arc to v , then av»v' is the only entry that is different from
A(D) in (i) above, and it now becomes a 1. This 1 is now a problem because it is
below the 0’s in column v , which had previously been labeled as C’s. So now move row v" just below the lowest row in A(D — v") for which there is a 1 in column v . Because v and v” do not have directed arcs to vertices in D*, av»v< is the only entry in row v that is a 1, so the 0 to the left (in column v") is still labeled as a C, and all the 0’s to the right can be labeled as R’s. Now A(D) has a zero partition.
(iii) If v has a directed arc to v", then av'v" is the only entry that is different from
A(D) in (i) above, and it now becomes a 1. This 1 is now a problem because it is
below the 0’s in column v”, which had previously been labeled as C’s. Now move row v to the top. Because v and v do not have directed arcs to vertices in D*, av'v» is the only entry in row v that is a 1, so all the other entries are 0’s, which are all to the right of entry av'v» and can each be labeled R. Now we have a zero partition for A{D).
In every case above we have found a zero partition of A(D), so Theorem 2.1.1 implies that D is an interval digraph.
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It is worth mentioning here that if there is no arc directed between v and v" then D can be proved to be an interval digraph with fewer restrictions placed on the directed arcs between the vertices of D* and v , v". The following theorem proves this fact.
Theorem 2.3.2 Let D be a directed graph on n vertices with no 2-cycles or loops, and with a transitive (n — 2)-subtournament D* = D-{v , v'}, where v and v" are vertices that have no arc directed between them in D.
Suppose v beats a subset of the vertices in D* that v" beats, or v" beats a subset of the vertices in D* that v beats. Also suppose that v is beaten by a subset of the vertices in D* that beat v", or v" is beaten by a subset of the vertices in D* that beat v .
Then D is an interval digraph.
Proof:
Because D is a digraph with no 2-cycles or loops that has a transitive (n — 2)-subtournament D*=D-{v , v"}, we know that the subgraph D-v" has a transitive subtournament on all but one of its vertices. Theorem 2.2.1 implies that D-v" is an interval digraph, which is equivalent to A(D — v") having a zero-partition from Theorem 2.1.1.
Arrange the submatrix A(D — v") as in the proof of Theorem 2.2.1, where the rows corresponding to vertices from D* with arcs directed to v are at the top, so that all the 0’s in column v are below the l’s and each is labeled as C.
If the vertices in D* with directed arcs to v" are a subset of those with directed arcs to v , then we can place column v" to the left of column v in A(D) and rearrange the top rows with l’s in column v so that all the rows with l’s in column v" are on top, with the 0’s underneath them labeled as C’s. (Note that column v will still have all its l’s at the top, with 0’s underneath and labeled as C’s).
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If the vertices in D* with arcs directed to v are a subset of those with arcs directed to v", then we can place column v to the left of column v again. All the rows with Is in column v are already at the top, which also have l’s in column v”, and we can place the other rows with l’s in column v" directly below these rows so that columns v and v" have the 0’s labeled as C’s and placed below all of the l’s.
Since we have arranged A(D — v") as in the proof of Theorem 2.2.1, row v is at the bottom of A(D — v") and has each of its 0’s labeled as C.
If the vertices in D* with arcs directed from v are a subset of those with directed arcs from v , then we can place row v" below row v in A(D) and label each of its 0’s as a C, since each of these C’s will either be below a 1 or another C.
If the vertices in D* with arcs directed from v are a subset of those with arcs directed from v", then we can place row v" just above row v in A(D) and label each of its 0’s as a C, since each of these C’s will always be above another C.
As in the proof of Theorem 2.2.1, the rest of the 0’s (which are from the submatrix A(D*) are labeled as R’s. Therefore A(D) has a zero partition, and from Theorem
2.1.1 this implies that D is an interval digraph.
â– 
Theorem 2.3.2 will fail to hold if there exist an arc between v and v". The figure
2.14 shows an induced 6-cycle in B(D) corresponding to the digraph D which has arc v i ^ v*. In D, v” beats a subset of the vertices in D- [v ,v"] that v beats and v is beaten by a subset of the vertices in T)-[v ,v'} that beat v .
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2.4 More than one nontrivial strong component
A strong component of a digraph D is a maximal strongly connected subgraph.
A directed graph is strongly connected if for every pair of vertices v, u there exists at least one path from v to u, and at least one path from u to v. Brown, Busch, Lundgren [6] proved the following theorem about strong components in tournaments. Theorem 2.4.1 No tournament that has two or more non-trivial strong components is an interval tournament.
However, an interval digraph can have more than one nontrivial strong component. In fact, certain digraphs with two or more nontrivial strong components that have specific types of adjacency matrices always have an interval representation, as the following theorem shows. Figure 2.15 shows an interval digraph with 2 non-trivial strong components. We will require a couple of new definitions to prove that certain digraphs with specific adjacency matrices and two or more non-trivial strong components will always be interval.
Definition 2.4.2 An R zero partition of a {0, 1 }- matrix is a zero partition in which each of the 0’s of the matrix can be labeled as R (where to the right of each R is another R) after column permutations. Likewise, a C zero partition is a zero partition in which each of the 0’s can be labeled as C (where below each C is another C) after row permutations.
The following theorem shows that a digraph with exactly two non-trivial strong components will be interval if the adjacency matrices of the strong components and the adjacency matrix determining the relation between the strong components are specific. It is worth mentioning here that the following result has been generalized to a digraph with k E N non-trivial strong components in [16].
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Theorem 2.4.3 Let D be a directed graph on n vertices with exactly two nontrivial strong components, Di and D2. Suppose that a subset of the vertices of D\ have arcs directed to a subset of the vertices of D2 (but no vertices in D2 have arcs directed to vertices in D\.
If the submatrices Ai = A(Di) and A2 = A(D2) both have a C zero partition, and the submatrix B (whose rows represent the vertices of D1; and whose columns represent the vertices of D2) has an R zero partition, then D is an interval digraph.
Proof:
Let the adjacency matrix A(D) be arranged as follows:
A(D) =
A B
yOA2j
Our hypothesis states that not only does D contain exactly two nontrivial strong components Di and D2, but their adjacency matrices, Ai and A2, must have a C zero partition after row permutations. Also, the arcs directed from vertices in Di to vertices in D2 must be such that the adjacency sub matrix B (whose rows represent the vertices of ZR, and whose columns represent the vertices of D2) must have an R zero partition after column permutations. Assuming that these conditions are met, the column permutations used to get an R zero partition of B (which do not affect the order of the entries in the columns of A\ or A2) will not change the C labelings of the Os in A\ and A2, and the row permutations used to get a C zero partition of A\ and A2 (which do not affect the order of the entries in the rows of B) will not change the R labelings of the Os in B. Therefore A(D) has a zero partition in which each of the Os in the submatrices represented by A\, A2, and 0 are labeled as C, and the
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submatrix B has each of its Os labeled as R. This implies (by Theorem 2.1.1) that D is an interval digraph.
â– 
We have studied the structures of several digraphs in order to ascertain when they are interval. We focussed on digraphs that possess some similar qualities of interval tournaments such as the digraphs with a transitive subtournament on all but two vertices and the digraphs for which the removal of one vertex leaves a subdigraph that is a transitive sub-tournament. These types of digraphs seemed to be the natural place to start because there is a complete characterization of interval tournaments. We successfully found different restrictions that force a digraph to be interval thus finding some special classes of interval digraphs.
There are still numerous classes of digraphs that will be interval with some restrictions on their arcs which have not been explored yet. Surely the problem of finding a complete forbidden subdigraph characterization of interval digraphs will be a very difficult one since the list of forbidden structures is already very long. Much work still needs to be done on this topic.
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No vertices from D* beat v' or v" and v" beats a subset of vertices that v' beats
v" has a directed arc to v'
ZERO-PARTITION
No directed arc between v" and v'
ZERO-PARTITION
Figure 2.12: Example of a digraph satisfying a condition of the theorem 2.3.1 and its zero partition
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v" beaten by a subset of vertices in D* that beat v' No directed arc between v" and v'
v2,vl—/ V v'— —7- v3
Vl 71 V" v'— —^ v2, v3
v" to the left of v', rearrange so that all l's in col v" are the top label 0's as C in the colv"
v" v' v3 v2 vl
v" v' v3 v2 vl
v2
vl
since vertices D* beaten by v' is a subset of vertices in D* beaten by v", so place row v" above v'
v2
vl
L
v3
v2
vl
v'
v"
0 0 0
Oil 1 1 1
0 0 1
0 0 1
vl
/ v3
v"
v'
1 1 1
C 1 1
C C R C C 1
c c 1
1 R
R R R R
1 C
C C
ZERO-PARTITION
Figure 2.13: Example of a digraph satisfying a condition of the theorem 2.3.2 and its zero partition
B(D)
induced 6-cycle from B(D)
Figure 2.14: Example of a digraph that fails the hypothesis of theorem 2.3.2 since v beats v and hence forms a 6-cycle in its bigraph representation
Figure 2.15: Strong components
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3. Interval Bigraph Impropriety
3.1 Introduction
An interval graph is proper if and only if it has a representation in which no interval contains another. Beyerl and Jamison introduced the study of p-improper interval graphs where no interval contains more than p other intervals in 2008. Thus a proper interval graph is a 0-improper interval graph. In this chapter we extend the idea by introducing p-improper interval bigraphs, where no interval contains more than p other intervals of the same partite set. Several authors have studied proper interval bigraphs. One of these characterizations has three forbidden subgraphs. We find bounds on the structure of p-improper interval bigraphs and characterize a special case of p-improper interval bigraphs.
Let a graph G have vertex set V and edge set E. x,y E V being adjacent is denoted by xy G E. A finite simple graph G(V, E) is an interval graph if we can find a mapping 6 : v —> Iv from vertices of G to intervals on the real line, such that the edge xy exists if and only if Ix n Iy ^ for all x, y G V(G). Interval graphs were first discussed by Hajos [22], The study of interval graphs also has its origin in a paper of Benzer (1959) [1] who was studying the structure of bacterial genes.
A well-known characterization of interval graphs was given by Lekkerkerker and Boland in 1962. The Lekkerkerker-Boland Theorem [26] says that chordless cycles and asteroidal triples form a defining class of forbidden subgraphs for the class of interval graphs. An asteroidal triple (AT) in G is a set A of three vertices such that between any two vertices in A there is a path between them that avoids all neighbors of the third. A natural extension of AT is an Asteroidal Triple of Edges. An asteroidal triple of edges (ATE) is a set of three edges such that for any two there is a path from the vertex set of one to the vertex set of the other that avoids the neighborhood of the third edge. A natural extension of intervals graphs, called interval bigraphs, were introduced by Harary, Kabell, and McMorris [23] in 1982. A bipartite graph
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G(X,Y,E) is an interval bigraph if to every vertex, v G V(G), we can assign an interval of the real line, Iv, such that xy G E(G) if and only if Ix n Iy ^ 0 and x G X and y eY. These graphs have been studied by several authors [13],[27]. To date no forbidden subgraph characterization of interval bigraphs has been found, but initially it was thought that asteriodal triples of edges along with induced cycles larger than 4 would work [23]. They proved that if B is an interval bigraph then B does not have ATE. However, Muller [31] found insects and Hell and Huang [24] found edge asteriods and bugs as forbidden subgraphs, and to date a complete characterization is still not available.
Fred Roberts [35] in 1969 characterized proper interval graphs. Proper interval graphs are graphs which have an interval representation such that no interval contains another. An interval graph is proper if and only if it does not contain as an induced subgraph. Unit interval graphs are the graphs having an interval representation in which all the intervals have the same length. Claw-free interval graphs are the interval graphs without an induced copy of the claw, Ki;3. Roberts has also shown that the class of proper interval graphs coincide with the classes of unit interval graphs and claw-free interval graphs.
Proper interval bigraphs are bigraphs which have an interval representations where no interval contains another. Several charactizations of proper interval bigraphs have been found in the last decade. The graphs in Figure 3.4 are the forbidden subgraphs for proper interval bigraphs found by Lin and West [27].
The idea of proper interval graphs was naturally extended to p-proper interval graphs by Proskurowski and Telle [33]. The p-proper interval graphs are graphs which have an interval representation where no interval is properly contained in more than p other intervals. Beyerl and Jamison [3] investigated a variation in containment and introduced p-improper interval graphs where no interval contains more than p other intervals. A p-improper representation of a graph is an interval representation of
28


the graph where no interval contains more than p other intervals. Figure 3.2 is an illustration of a 1-improper interval graph. Thus any 0-improper interval graph is a proper interval graph. It can be easily checked in Figure 3.1 that Ah,3 is a 1-improper interval graph.
In this chapter we generalize the class of p-improper interval graphs to p-improper interval bigraphs. We look at ideas which Beyerl and Jamison investigated for interval graphs and apply them in interval bigraphs. Some definitions had to be modified since we are working with two partite sets.
A p-improper interval bigraph is an interval bigraph where no interval contains more than p intervals from the same partite set. In order to be able to generalize the ideas for p-improper interval graphs to p-improper interval bigraphs, we restrict the containment of intervals to an individual partite set as shown in the Figure 3.3. Beyerl and Jamison investigated ideas such as impropriety, weight, p-critical and balance for interval graphs, and we investigate the same ideas with some modified definitions for interval bigraphs. In this chapter we find restrictions in the stucture of an interval bigraph for it to be a p-improper interval bigraph. We also study special classes of p-improper interval bigraphs.
Figure 3.1: Ad,3
3.2 Impropriety and Weight of Interval Bigraphs
From this point on B = B(X,Y, E) will denote a finite, connected, interval bigraph with bipartition {X, Y} and the sets X and Y will be referred to as the partite
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p-improper interval graph (here p=l)
No Interval contains more than p other intervals
(By symmetry representatiom with b ,d as base are same as that of c)
Figure 3.2: p-Improper Interval Graph
Figure 3.3: O-Improper Interval Bigraph
sets. A 0-improper interval bigraph is a proper interval bigraph. The following proposition justifies our way of defining a p-improper interval bigraph. It explains the sufficiency of restricting the inclusion of intervals to a single partite set in an interval bigraph.
Proposition 3.2.1 B is a proper interval bigraph if and only if it has an interval representation where no interval from a partite set contains another interval from the same partite set.
Proof: Suppose B(X,Y,E) is a proper interval bigraph. It follows from the
definition that it has an interval representation where no interval from a partite set contains another interval from the same partite set, since no interval contains another.
Suppose B has an interval representation where no interval from a partite set
30


contains another interval from the same partite set. If B is not proper then it has one of the graphs in Figure 3.4 as an induced subgraph [27]. It is easy to check from the interval representation of Bl, B2 and B3 (illustrated in Figure 3.4 and Figure 3.5) that they all are forced to have an interval which contains another interval from the same partite set. In Figure 3.4 we avoid the containment of an interval from Y in another interval from Y. This forces the containment of an interval from X in another interval from X. In Figure 3.5 we avoid the containment of an interval from X in another interval from X. This forces the containment of an interval from Y or X in another interval from Y or X. Bl is the only example where we never attain a containment from Y. It always has an interval from X containing another from X. Thus we cannot avoid an X interval properly containing another from X or a Y interval properly containing another from Y. Hence B has an interval which contains another interval from the same partite set which contradicts our assumption. â– 
BI(X.Y) B2(X,Y) B3(X,Y)
—--
Figure 3.4: Forbidden subgraphs for proper Interval bigraph with their interval representations to force containment from X
x2
y2
y2
xl
"77"
y3
x4 xl
—a
V2
I I
yi
y3
i i
x2
yi
y3
W
Figure 3.5: Interval representations of Bl, B2, B3 to force containment from Y
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First we will generalize the notion of impropriety introduced by Beyerl and Jamison. The idea of impropriety comes from p-improper interval bigraphs.
Definition 3.2.2 The impropriety of a vertex z E X with respect to the interval representation a is the number of vertices x E X such that Ix C IZJ which we will denote as impa(z).
Definition 3.2.3 The impropriety of a representation a is the maximum impa(z) for all z E X,Y which we will denote as imps (a).
Definition 3.2.4 The impropriety of the interval bigraph B is the minimum impsia) over all possible interval representations which we will denote as imp(B). A representation which gives the minimum impropriety will be called a minimal representation.
Any interval bigraph can have an infinite number of interval representations. So it is easy to see that the impropriety is the least p for which the graph is a p-improper interval bigraph as shown in the Figure 3.6. Figures 3.8 and 3.9 give examples of graphs whose impropriety is 1 and 2 respectively. It is easy to check that Iz D Ix,Iy is forced.
Definition 3.2.5 For z E B(X,Y, E), a component of B — z will be called a local component of z. A local component of z is exterior if and only if it contains an edge xy such thatx,y N(z). (See Figure 3.7).
It is worth mentioning here that all the bigraphs here have connectivity one.
The following lemma gives us a bound on the number of exterior local components any vertex of an interval bigraph can have.
Lemma 3.2.6 A vertex z in an interval bigraph can have at most two exterior local components.
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MINIMAL REPRESENTATION
Figure 3.6: Impropriety of imp(B(f>1) = 2, imp(Bfa) = 1
Figure 3.7: Exterior local components
Proof: We will prove this by contradiction. Let us assume that B(X,Y) is a connected interval bigraph and ; E X is a vertex in B with at least 3 exterior local components. So there exists 3 edges xpyi, #22/2 , #32/3 hi 3 of these exterior components such that #1, yi, #2i 2/2) #3i 2/3 ^ tV(T). We can find a path between any two pairs of these edges, say from xpyi to #22/2 through c which avoids the neighborhood of #32/3 since #3,2/3 4- N(z). Hence the edges #r2/i,#22/2)#32/3 form an asteroidal triple of edges which is a forbidden subgraph for an interval bigraph [23]. Thus we get a contradiction. Hence any vertex c in an interval bigraph can have at most two exterior local components.
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Note: Let B(X, Y) be an interval bigraph. If C% is an exterior component of any vertex z E X then in the interval representation there exist a vertex y E Ci fl Y such
Figure 3.8: Interval bigraph with impropriety 1
Figure 3.9: Interval bigraph with impropriety 2
The following definition and theorem leads to a bound on the impropriety.
Definition 3.2.7 Let B(X, Y, E be an interval bigraph. Let z E X and C\,C-2,...., Cn be the local components of z ordered from left to right (the left hand components have intervals on the left hand side of the interval representation and the right hand components will have intervals on the right hand side of the interval representation) such that C\ and Cn are chosen either to be exterior components if they exist or non-exterior local components such that \C\ fl X\ > \Ci fl X|,?' = 2,— 1) and
\Cn fl X\ > \Ci fl X|, i = 2,— 1). The weight of the vertex c is the total number
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of vertices in (C-2 fl X) U .... U (Cn-1 fl X), which we will denote as wt(z). We will call C2, C3,...., Qn-i) the smaller non-exterior local components of z. The weight of X is the maximum of the weights of all the vertices x E X, which we will denote as wt(X). The weight of a vertex in Y and the weight ofY are defined similarly. The weight of an interval bigraph, wt(B), is the maximum of the weights of its two partite sets.
The graphs in Figure 3.9 and Figure 3.11 have weights 2 even though they have a different number of exterior components. This is because we consider the smallest non-exterior local components to calculate the weights.
(n-2)- components
C3 Cn-1
Figure 3.10: Calculation of weight
Figure 3.11: Interval bigraph with weight 2
The next theorem gives us a bound on the impropriety.
Theorem 3.2.8 Let B(X,Y,E) be an interval bigraph. Let z E B. Then the impropriety of B is at least the weight of z.
Proof: Let a be a minimal interval representation of the interval bigraph B.
Let z E X and C\, C2,...Cn be the local components of c ordered from left to right
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as in Definition 3.2.7. Let Iz denote the interval of z. So wt(z) = JT=2 (ra_i) \CiC\X\.
The intervals of vertices from X in C2,.., Cn-1 must be inside Iz because there are
vertices y\ E C\, yn E Cn that are not adjacent to any vertex from X in C2,., Cn-1
but are adjacent to z. Hence, for any a and for any z E X wt(z) < imp(z). Since imp(B)=max[ imp(zi) for all z* G B ], irnp(z) < irrip(B), and so wt(z) < irrip(B). Hence the impropriety of B is at least the weight of z.
Corollary 3.2.9 For any interval bigraph B(X,Y,E), imp(B) > wt(B).
Proof: The proof follows directly from the previous theorem and definitions, m
Note: Now that we have the above bound on weight, the reason behind the restriction of the containment to a single partite set can be discussed more explicitly. If we did not confine it to an individual partite set, the above result would fail to hold for interval bigraphs. For example, consider K^4. This is an interval bigraph. Let us call it B(X,Y, E). Without loss of generality, there is only one vertex in X and the remaining four vertices belong to Y. By our definition of impropriety and weight, we have the wt(X) = 0 = imp(B). Thus the above bound wt(X) < imp(B) holds here. If we had not restricted the impropriety to a single partite set then the definition of weight would also be different. Weight also would not have been confined to an individual partite set. It would take into account all the vertices in the n — 2 smallest non exterior components of any vertex with n local components. Therefore imp(B) = 0 and wt(X) = 2 which is illustrated in Figure 3.12. This defies the bound wt(X) < imp(B). This bound holds for interval graphs and since it is easier to calculate wt(B)than imp(B), we wanted this useful bound here.
3.3 p-critical interval bigraphs
We have seen in corollary 2.9 that the weight of any partite set of an interval
bigraph can be at most the impropriety of the bigraph. Figure 3.13 and figure 3.14
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Figure 3.13: Interval bigraph where wtfX) = 0 < imp(B) = 1
demonstrate the two cases. We will look at the case when equality holds. The case when imp(B) = wt(X) will lead us to the following definitions.
Definition 3.3.1 Let B(X,Y,E) be an interval bigraph. X is balanced if and only if wt(X) = imp(B). If X is balanced, there exists a vertex ; 6 I such that wtfz) = imp(B) and such a vertex is called a basepoint of X.
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Figure 3.14: Interval bigraph where wtfX) = 1 = imp(B) = 1
Note that in any balanced partite set of an interval bigraph with imp(B) > 0 the basepoint must have at least 3 local components. Otherwise the weight becomes 0 and hence the impropriety is 0, leading to a contradiction. Moreover if the wt(X) > 0 then the basepoint of X must have at least 3 local components with vertices from X in all three components.
Definition 3.3.2 A partite set X of an interval bigraph B(X,Y,E) is p-critical with respect to impropriety if and only if B has impropriety p, and the removal of any vertex z E X decreases the impropriety of B.
The balance and criticality of different partite sets might not be the same for an interval bigraph. In Figure 3.15 we have illustrations of balance and criticality of the partite sets of various interval bigraphs. We prove later in this section that both the partite sets of an interval bigraph cannot be balanced and p-critical at the same time.
We will now focus our attention on a special class of graphs. For the remaining part of this section we will consider interval bigraphs with a balanced and p-critical partite set. With this restriction on a partite set of an interval bigraph we will prove interesting results about the structures of this particular class of bigraphs. The next theorems and lemmas give us an idea about the structure of an interval bigraph with a balanced and p-critical partite set.
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y>
y
Al(ii)
Xis not balanced , 1-critical Y is not balanced , 1-critical
X is balanced , not lcritical Y is balanced ,not 1-critical
A2(i)
A4(i)
A4(ii)
X is balanced , 1-critical Yis not balanced , not 1-critical
Figure 3.15: Illustrations of balance and p-criticality
Theorem 3.3.3 Let z be a basepoint of a balanced p-critical partite set X of the Bigraph B(X,Y). If C is an exterior local component of z, then C has exactly one vertex from X.
Proof: Let B(X,Y) be an interval bigraph. Suppose X is balanced and p-
critical and c is its basepoint. We will prove this by contradiction. We use techniques
similar to those used by Beyerl and Jamison [3]. Let C be an exterior local component
of c that has at least 2 vertices x,v E X. Let x be the closest vertex to c such that
it is on an edge xy and y N(z). Figure 3.16 illustrates possible positions of the
vertex x. Let H be the graph obtained from B(X,Y) by deleting all the vertices
from C except those on the path from c to y that contains x. By theorem 3.2.8
< imp(H). Since X is p-critical, imp(H) < imp(B). Again, since X is
balanced imp(B) = wt(X). Since c is a basepoint of X, wtx(z) = wt(X). Since
the vertices removed from X are in the exterior component C, the smallest (n — 2)
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non-exterior components remain the same, which implies wtx(z) = wtn(z)-
Hence wtn(^) < imp(H) < imp(B) = wt(X) = wtx(z) = wtn(z), which is a
contradiction.
Figure 3.16: Choice
Next we show that if a partite set X is balanced and p-critical then it has a unique basepoint. Then we will prove several facts about interval bigraphs using the unique basepoint of either of its partite sets. It is easy to see that if the weight of any partite set X of an interval bigraph is 0 then a basepoint c has at most 2 local components with vertices from X. If the wt(X) > 0 then a basepoint has at least 3 local components with vertices from X in all three components. Depending on different scenarios, the local components can be at the side of the interval Iz or must be totally contained in it. In any interval representation, we will call the leftmost and rightmost end local components of any vertex its local side components. The remaining non-side local components will be called the local inner components. In Definition 3.2.7, C\ and Cn are always the local side components. If they are exterior local components then they will contain vertices from Y non-adjacent to c and so they must be on the sides of Iz. If they are not exterior then in order to obtain a minimal representation they must again be on the two sides of Iz. We will call C\ the leftmost local component and Cn the rightmost local component.
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Lemma 3.3.4 Any balanced and p-critical (p > 0) partite set of an interval bigraph B(X,Y,E) has exactly one basepoint.
Proof: Let X be a balanced and p-critical partite set of an interval bigraph B(X,Y, E). We need to show that X has only one basepoint. We will prove this by contradiction. Let us assume that X has two basepoints namely x and v. Since X is balanced and p-critical (p > 0), both x and v must have at least 3 local components with vertices from X. Since x is a basepoint, it has at least 3 local components and v must be in one of them. This leads to the following cases:
Casel : If v G C where C is an exterior local component of x, then, by Theorem 3.3.3, v is the only vertex from X in C. This implies that v can have exactly one local component containing vertices from X. This is a contradiction.
Case2 : If v G C where C is a non-exterior local inner component of x then every vertex from Y which is adjacent to v is also adjacent to x since C is not exterior. Thus B-v is connected. So v has just one local component which is a contradiction. Case3 : If v G C where C is a non-exterior local side component of x, every vertex from Y which is adjacent to v is also adjacent to x because C is not exterior. We can conclude that v again has just one local component, which is a contradiction.
â– 
The following theorem proves that both the partite sets of an interval bigraph cannot be balanced and p-critical at the same time.
Theorem 3.3.5 Let B(X, Y) be an interval bigraph. Both X and Y cannot be balanced and p-critical p > 0 at the same time.
Proof:
We will prove this by contradiction. Let us assume that both X and Y are balanced and p-critical. Let v and w be the basepoints of X and Y respectively. Since X is balanced and p-critical and p > 0, its basepoint must have at least 3 local
41


components with vertices from X. Let C\, C2, ■ ■Cj, ....Cn be n local components of v ordered from left to right and n > 3. Hence v is adjacent to at least n vertices from Y. Since B is connected, w must be in some local component of v. Let w E Ck, k E 1, 2, ...n. Since Y is balanced and p-critical, w must have at least 3 local components with one local component containing ( v U Cf), i = 1, ...n and i ^ k as an induced subgraph, which has at least (n — 1) vertices from Y. Let us call this local component of w Ai where KH4 nT)| > (n— 1). Since n > 3 there are at least 2 local components of v say Cj, Cm and j, m ^ k, each of which contains a vertex from X not adjacent to w. Since Cj and Cm are contained in Ai, it implies that Ai is an exterior component of w. Since Ai is an exterior component of w it can contain exactly one vertex from Y [Theorem 3.3.3]. But |(Hi nh)| > (n — 1) and n > 3 which is a contradiction. Hence both X and Y cannot be balanced and p-critical at the same time.
â– 
Next we determine the general structure of an interval bigraph that has a balanced and p-critical partite set. The following lemma and theorem give an explicit idea about the cardinality of the local side components of the basepoint of any balanced and p-critical partite set of an interval bigraph. The lemma gives us the cardinality and the following theorem uses the lemma to demonstrate the structure. If a representation has been given, lv and rv will denote the left and right endpoints, respectively, of the interval Iv representing any vertex v.
Lemma 3.3.6 Let X be a balanced and p-critical partite set of an interval bigraph
B(X,Y) and v G X be the basepoint of X with at most one exterior component. Let
Ci,C2,.....Cn be local components of v ordered from left to right as in Definition
3.2.7. Let a be a minimal interval representation such that C\ is a non-exterior local
side component of v in a and Cn is the other local side component. If Cj is a local
inner component of v such that \Cj fl X\ is largest among all the other local inner
components ofv, then IC^-nXl = |C'iflX| if Cn is exterior and IC^flXl = |C'iflX|
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= |Cn n X| if Cn is non-exterior.
Proof:
We will prove this by contradiction. Let v G X be the basepoint of X with the corresponding minimal interval representation a. The local components of v with
respect to the minimal interval representation a are C\, C2,.....Cn ordered from left
to right with C\ as the non- exterior local side component and Cn as the other local side component. First let Cn be an exterior local component and pick j such that \Cj fl X| > l^nxi for all i = 2,3, ....(n — 1).
From Definition 3.2.7, \C\ fl X\ > \Cj fl X\. Assume that |C\ fl X\ > \Cj fl X\, then |Ci fl X\ > \Ci fl X\, i = 2, 3, ....(n — 1). Let X\ G C\ fl X and look at B — x\. Since C\ is non-exterior, every vertex from Y in C\ adjacent to X\ is adjacent to v. So B — x\ is still connected. Since \C\ fl X\ > \Ci fl X\, i = 2, 3, ....(n — 1), removal of x\ from C\ does not affect the weight of X. Hence wt(X — x\) = wt(X). Since X is p-critical, removal of any vertex from X will reduce the impropriety of B by one. Thus imp(B — x\) < imp(B) and X being balanced implies imp(B) = wt(X). So we have the following relation: imp(B — x\) < imp(B) = wt(X) = wt(X — x\).
Thus imp(B — x\) < wt(B — x\), which is a contradiction. Hence \Cj fl X\ =
|Ci n x| .
If Cn is also a non-exterior local side component then by the same argument we can prove that |Cj fl X\ = \Cn fl X\. Hence |Cj fl X\ = \Cn fl X\ = |C\ fl X\.
â– 
The next theorem illustrates the structure of an interval bigraph B with a balanced and p-critical partite set where p > 0. It proves that B can have exactly three structures depending on the number of exterior components.
Theorem 3.3.7 Let B(X,Y, E) be an interval bigraph. If X is balanced and p-critical
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1 exterior components
—
y5 y7
yi___ y2 y^_ ______________y4
v
2 exterior components
Figure 3.17: Possible structures of an interval bigraph with a balanced and p-critical partite set
then B has one of the following structures:
i) B ~ B\ ■where B\(X,Y) is an interval bigraph and v is the basepoint of X which has n local non-exterior components Ci,C-2, ..Ci, ....Cn ordered from left to right as in Definition 3.2.7 such that there are at least 3 local components Ci,Cj,Cn •where
|Cj fl X\ = |C\ Pi X\ = \Cn n X\ > \Ci n X|, i = 2,.., ('/?, — 1). Furthermore each of
Ci and Cn has a vertex from Y which is adjacent to all vertices from X in C\ and Cn .
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ii) B ~ B2 where B2(X,Y) is an interval bigraph and v is the basepoint of X vjhich has exactly one exterior component Cn and (n — 1) local non-exterior components Ci, C2, ■ ■Ci, ....Cn-1 ordered from left to right as in Definition 3.2.7 such that it has at
least 2 local components C\, Cj where \CjC\X\ = |67i6lX| > |67j6lX|, i = 2,.(n— 1).
Furthermore the left most local component C\ has a vertex from Y which is adjacent to all vertices from X in C\.
iii) B ~ B3 where B3(X,Y) is an interval bigraph and v is the basepoint of X which has Ci, C2, ..Ci, ....Cn local components ordered from left to right with exactly two exterior components Ci,Cn and the remaining (n — 2) components are local inner components.
Proof: B(X,Y, E) is an interval bigraph where X is balanced and p-critical. We need to prove that B is isomorphic to Bi,B2 or B3 depending on the number of exterior components the basepoint of X has. Let v G X be the basepoint of X with n local components Ci,C2, ..Ci, ....Cn ordered from left to right. By Lemma 3.2.6 we know that v can have at most 2 exterior components and this gives us 3 cases.
Case I: If v G X has no exterior components then all the components of v are nonexterior local components. Hence by Lemma 3.3.6 there must be at least 3 components
Ci,Cj,Cn where \Cj f] X\ = \C\nX\ = |67ra6lX| > |67;6lX \,i = 2,....(n - 1). Thus
B ~ Bi.
Let us now analyze the local side component Ci. Since Ci is not an exterior component of v G X, it can have more than one vertex from X. Since the weight of v is determined by the smallest (n — 2) local inner components of v and by the previous Lemma 167161X1 = \CjC\X\ > 167161X1 for all i = 2, 3, ....(n — 1), the component Ci does not contribute anything towards the weight of v. Let us assume that \Cj 61X| = k. Hence |67i 61 X\ = k. Since X is balanced, p-critical with p > 0, Ci must have vertices from
45


both X and Y. Let X\,X2,.......Xk G C\. There can be more than one vertex from Y
in C\. Since C\ is a non-exterior component, Iyj fl Iv yt for all yj G C\ (*).
Since X is balanced, no Xi contributes to the impropriety of v for Xi G C\, which implies that none of the ay, i = 1,2, ....n are contained in Iv(**).
Since C\ is the leftmost local component, it implies that
lXi < lv from(*) for all i ryj ^ lv. from(**) for all j
Take the ay such that rXi is the least. This ay must be adjacent to some yj which is adjacent to v. Thus Iyj fl IXi yt (pi and by our choice of ay, Iyj fl IXm yt

Case II: If v G X has exactly 1 exterior component then it has a non-exterior side component. Let Cn be the exterior component. Again by Lemma 3.3.6 it must have at least 2 focal components Cj and C\ such that \C\ fl X\ = \CjC\X\ > \Ci fl X\ for all i = 2...., (n — 1) and one of these, say C\, must be the focal side component. Thus B ~ £>2. By arguments similar to those used in Case I, C\ also has a vertex from Y which is adjacent to all vertices from X inC\. This is illustrated in Figure 3.17.
Case III: If v G X has 2 exterior components C\ and Cn then all the other (n— 2) focal components of v are non-exterior local components. The exterior focal components cannot be focal inner components otherwise we would not be able to obtain any interval representation of the bigraph. Hence the two exterior components C\, Cn must be the two side components of v and the remaining (n — 2) components are the focal inner components. Thus B ~ B3. This structure is also illustrated in Figure 3.17.
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3.4 Conclusions and future work
We investigated the structures and characteristics of p-improper interval bigraphs. The same ideas could be naturally extended to interval fc-graphs. These are graphs with a proper coloring where each vertex v can be assigned an interval Iv of the real line such that two vertices are adjacent if and only if their corresponding intervals overlap and each vertex has a different color. It would be interesting to study the roles played by impropriety and criticality on interval fc-graphs.
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4. Characterization of unit probe interval 2-trees 4.1 Introduction
Interval graphs were introduced by Hajos [22], and were then characterized by the absence of induced cycles of length larger than 3 and asteroidal triples by Lekkerkerker and Boland [26] in 1962. Unit interval graphs are the interval graphs that have an interval representation in which each interval has unit length. In 1969, Roberts [36] proved that the classes of proper interval graphs and unit interval graphs coincide and he showed that interval graphs that have no induced Ad,3 are unit interval graphs. Much more recently in 2007 Gardi gave a much shorter new constructive proof of Roberts original characterization of unit interval graphs [18].
A graph is a probe interval graph if there is a partition of V(G) into sets P and N and a collection {Iv : v G V(G)} of intervals of R such that, for u,v G V(G), uv G E(G) if and only if Iu n Iv ^ and at least one of u or v belongs to P. The sets P and N are called the probes and nonprobes, respectively. If, for G a probe interval graph, the members of Iv : v G V(G) are closed intervals of identical length, then G is a unit probe interval graph. The probe interval graph model was invented in connection with the task called physical mapping used in connection with the human genome project by Zhang and Zhang et. al. [43], [44],
One way to describe the structure of a class of graphs is by finding its complete list of minimal forbidden induced subgraphs; No complete forbidden induced subgraph characterization for general probe interval graphs has been found so far. As the characterization of probe interval graphs seems to be difficult, research has focussed on classes of probe interval graphs.
Family of 2-trees are the set of all graphs that can be obtained by the following construction: (i) the 2-complete graph, K2, is a 2-tree; (ii) to a 2-tree Q' with n — 1 vertices (n > 2) add a new vertex adjacent to a 2-complete subgraph of Q'. Li Sheng first characterized cycle-free probe interval graphs [41]. As a natural extension of
48


the characterization for trees, Przulj and Corneil attempted a forbidden subgraph characterization of 2-trees that are probe interval graphs and they found at least 62 distinct minimal forbidden induced subgraphs for probe interval graphs that are 2-trees [34], More recently Brown, Flesch and Lundgren extended the list to 69 and gave a characterization in terms of sparse spiny interior 2-lobsters [7]. In 2009 Brown, Sheng and Lundgren gave a characterization of cycle-free unit probe interval graphs [15]. In [11] various characterizations are given for the probe interval graphs that are bipartite. These characterizations are for classes of bipartite graphs, but no general characterization of bipartite probe interval graphs were given there. However recently, Brown and Langley characterized unit probe interval graphs for bipartite graphs [10]. This chapter restricts to the unit case of probe interval graphs which are 2-trees.
In this chapter we characterize 2-trees that are unit probe interval graphs. In Section 2 we introduce some important subclasses of 2-trees. In addition, we introduce several basic results that will be useful in the characterization. In the Section 3 we characterize 2-caterpillars and interior 2-caterpillars in terms of forbidden induced subgraphs and show that 2-trees that are unit probe interval graphs have to be interior 2-caterpillars. While this significantly reduces our focus, it turns out the problem remains very difficult. A major reason for this is that even determining which 2-paths are unit probe interval graphs is difficult. So in Section 4 we give a list of forbidden subgraphs for 2-paths that are unit probe interval graphs. Then in Section 5 we complete the characterization for 2-paths which are unit probe interval graphs. In Section 6 we extend the list of forbidden subgraphs from section 4 to include the remaining forbidden subgraphs for interior 2-caterpillars. In Section 7 we complete the list of forbidden subgraphs for 2-tree unit probe interval graphs using 27 subgraphs. The size of the list illustrates the difficulty of this problem.
4.2 Preliminaries
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Figure 4.1: Examples of some 2-paths
In this section we introduce some subclasses of 2-trees. Also we give several basic results that will be useful throughout the chapter. To describe the structure of 2-trees, we use the idea of a 2-path introduced by Beineke and Pippert in [2],
Definition 4.2.1 [2] A 2-path is an alternating sequence of distinct 2 and 'i-cliques, (e0, t\, e\,t-2, e-2,...,tp, ep), starting and ending with a 2-clique and such that U contains exactly two distinct 2-cliques e^_i and e* (1 < i < p). The length of 2-path is the number, p, of 3-cliques.
In general 2-paths are much more complex than paths. As we will see later that characterization for 2-paths which are unit probe interval graphs is very difficult.
Drawing the connection to tree further, Proskurowski introduced the notion of a 2-caterpillar in [32]. A 2-leaf is a vertex whose neighborhood is a 2-clique.
Definition 4.2.2 [32] A 2-caterpillar P is a 2-tree in which the deletion of all 2-leaves results in a 2-path, called the body of P. A 2-caterpillar P is an interior 2-caterpillar if for any 2-leaf v, v is adjacent to all vertices of some 2-complete subgraph e* of any longest 2-path of P.
In the Section 3 we will obtain a characterization of 2-caterpillars and interior 2-caterpillars.
Definition 4.2.3 The 2-distance between a 2-leaf and a longest 2-path of G is the length of the shortest 2-path between them.
Definition 4.2.4 An asteroidal triple (AT) in G is a set A of three vertices such
50


2-caterpillar interior 2-caterpillar
violates interior
Figure 4.2: On the left is a 2-caterpillar, and on the right is an interior 2-caterpillar
that between any two vertices in A there is a path that avoids all neighbors of the third.
It is well-known that interval graphs cannot contain asteroidal triples. However, probe interval graphs can contain asteroidal triples provided the restrictions in the following lemmas are obeyed. The following lemmas will also be very useful in proving some of the results in the present and forthcoming sections specially in showing that certain 2-trees that are probe interval graphs are not unit probe interval graphs.
Lemma 4.2.5 [fl] At least one vertex in an AT of a probe interval graph must be a non probe.
Lemma 4.2.6 [fl] In every AT there must exist a non-probe vertex u such that there exist a path between the other two vertices in the AT that has a non-probe internal vertex.
Lemma 4.2.7 [6] The vertex of degree 3 in any induced Ad,3 of a unit probe interval graph must be a probe and at least two of the vertices of degree 1 of the induced Ad,3 must be non-probes.
Definition 4.2.8 A fc-fan is a 2-path of length k such that all e* ’s are incident to a common vertex which we call the center; all other vertices are called the radial vertices.
Definition 4.2.9 f-fan is a 2-tree made up of four 2-clique.s such that all the cliques
share one particular vertex called the central vertex as in Figure f.3. The vertices
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along the circumference of the 4-fan are called the radial vertices. Radial vertices with degree 2 are the end radial vertices. The vertex with degree 3 which is equidistant from both the end radial vertices will be called the central vertex or mid radial vertex.
Definition 4.2.10 The Merge of two 2-trees Ti and Tj takes place if the intersection of their vertex set is not null. So ViTf) n V(Tf) ^ where V(Ti) and V(Tf) denotes the vertex sets ofTi and Tj respectively.
For convenience, we will introduce some notation. From now on the right end point and left end point of the interval corresponding to a vertex v will be denoted by r(Iv) and l(Iv) respectively. By m e t) we will mean adjacency between u and v. We will also use the acronym UPIG for unit probe interval graph.
We will name a 4-fan by a list of six vertices, say (a — b — c — d — e — f) with center at c by which we mean (a, b, d, e, /) is the path of radial vertices and c is the center of the 4-fan.
Lemma 4.2.11 The center of a 4-fan must be a probe, the rnid-radial (or the central vertex) vertex and at least one of the end-radial vertices must be non-probes; the other two vertices must be probes.
Proof:
Let F = ({ c, ri, f2, f3, f4, f5 },E) be a 4-fan where ri,., rs, in order, from the
path which contains the radial vertices and c is the center vertex. By Brown, Sheng, Lundgren, c is a probe and at most one of { ri, r3, r5 } is a probe and hence r2 and r4 must be probes. Now observe the representation Iri = Ir2 = [0,1], Ic = Ir3 = [1,2], Ir4 = Irh = [2, 3], works with ri, r5 as non-probes; if r5 is a probe ( or r\ is a probe), then Ir3 = [0.5,1.5](/^ = [1.5, 2.5]) can be used.
â– 
A 3-sun, denoted FI and depicted in the Figure 4.6 is a 2-caterpillar formed by
three 3-cliques with a 2-leaf on the non-interior edge of the 3-clique at the middle. It
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u2
Figure 4.3: 4-fan
U1
u3
u5
I u6
] U4
U2
Figure 4.4: Unit probe interval representation of a 4-fan
u2
U6
U3
Figure 4.5: An illustration where unit probe interval representation of a 4-fan fails to work
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U-i U3 U5
Figure 4.6: FI
is a well-known fact that the 3-sun is not an interval graph. It is easy to show that it is a probe interval graph. In the next lemma we will prove that it is not a unit probe interval graph.
Lemma 4.2.12 The 2-tree FI (3-sun) is not a unit probe interval graph.
Proof:
Let us assume that FI is a UPIG. It can be observed from Figure 4.7 that FI is a probe interval graph. Without loss of generality, by Lemma 4.2.5, let uq be a non probe. Since u2 and u4 are adjacent to uq, so they are probes. By Lemma 4.2.6, U3 must be a non probe. Hence u\ and U5 are probes. We now attempt to draw the unit probe representation of FI. We know that IU3 fl IUl yt 0 and IU3 fl IU6 yt 0. We make the intervals of U\ and u5 as far apart as possible. Thus IU3 is flanked on both sides by /M1and IU6. IU2 and IUi are drawn such that r(lui) < r(IU2) and r(IU4) < r(IU6) as shown in Figure 4.7. u6 is adjacent to u2 and u4 and the intervals of u2 and u4 overlap somewhere along the middle of the interval of U3 as shown in Figure 4.7. Hence the length of the interval of u6 must be reduced so that it can avoid overlapping with the intervals of u\ and U5 which is a contradiction. Therefore, Fl is not a unit probe
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[
]
U6
Ul —
U5
_________U4
U?
I -------------1
U3
Figure 4.7: Probe interval representation of a 3-sun or FI interval graph.
â– 
We will now prove that probe interval graph El shown in figure 4.8 is not a unit probe interval graph. We will later use these two results ( FI and FI do not have unit probe interval representations) to characterize interior 2-caterpillars.
Lemma 4.2.13 The 2-tree probe interval graph El shown in Figure 4-8 is not a unit probe interval graph.
Proof: We know that FI is a probe interval graph. Let us assume that FI has a unit probe interval representation. Vertices u2, Ui, u3, u4, u5 and u6 form a 4-fan as in Figure 4.8. The vertex u2 is the center of the 4-fan and so it must be a probe. The vertex u4 is the central radial vertex of the same 4-fan and so it must be a non-probe. As seen in Figure that u4,u3,Ci and u5 form an induced Ad,3 with center at u4 and so u4 must be a probe which is a contradiction. Hence FI is not a unit probe interval graph.
4.3 Characterization of 2-caterpillars
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u4
Figure 4.8: El
In this section we significantly reduce the class of 2-trees we need to consider for unit probe interval graphs. First we reduce the class of 2-tree unit probe interval graphs to 2-caterpillars and then to interior 2-caterpillars. Thus in this section we give a complete characterization of 2-caterpillars and interior 2-caterpillars. We will see that the complete list of forbidden subgraphs for 2-trees which are 2-caterpillars consists of 4 subgraphs. Furthermore we also prove that all 2-trees which are unit probe interval graphs must be interior 2-caterpillars. This result significantly reduces our spectrum from a wide range of graphs to a relatively small class of interior 2-caterpillars.
Theorem 4.3.1 A 2-tree is a 2-caterpillar if and only if it does not contain Bl, B1 B2, B3 as induced subgraphs.
Proof: Let G be a 2-tree which is a 2-caterpillar. By the definition of 2-
caterpillar, G is a 2-tree such that after deletion of all 2-leaves of G we get a 2-path,
called the body of P. If G= Bl, Bl', B‘2, B3 then G - {2-leaves of G } is not a 2-path
as seen in Figure 4.9 and so G cannot contain them as induced subgraphs.
Let G be a 2-tree without Bl, BY, B2, B3 as induced subgraphs. Suppose G is not
a 2-caterpillar. Let (e0, t\, e\, t-2, e-2, ...,tp,ep) be a longest 2-path of G and label it P.
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Since G is not a 2-caterpillar, G - {2-leaves of G } is not a 2-path. If every 2-leaf of G were a 2-leaf at distance 1 from P then G - { 2-leaves } would be a 2-path which is a contradiction. Hence G must have a 2-leaf at 2-distance 2 or more from P and this can happen in two ways. G has a 2-path of length at least 2 originating from some e* or it has a 2-path of length at least 2 originating from a pair of vertices of some T not equal to e*_i or e*
Case(l): Assume that G has a 2-path of length at least 2 originating from some e*. Since P is assumed to be a longest 2-path of G, there exists a 2-path of length at least 2 on either side of on P. Let X be a 2-path of length 2 which originates from 2-clique e*. Therefore the smallest structure satisfying these conditions is a 2-path P of length 4 with X starting from the middle 2-clique, e» and its length is 2. According to Przulj and Corneil [34], there are 2 non-isomorphic 2-paths of length 4 which are A\ and A% as in Figure 4.10. The first vertex x\ of X has 2 choices of positions as in Figure 4.11 and the second vertex :r2 of X has 2 choices again. Therefore we get four 2-tree subgraphs A*, % = 1,2, 3, 4 as in Figure 4.11. Observing that A*, % = 1,2,3, are isomorphic to B1 and A4 is isomorphic to B1', it follows that G cannot contain these 2-paths. Hence G cannot have 2-paths of length 2 or more originating from some interior edge e*.
Case(2): Now let us assume that G has a 2-path X of length at least 2 originating from a pair of vertices of some A not equal to e*_i or e*. Since P is assumed to be a longest 2-path of G, there exists a 2-path of length at least 2 on either side of A on P. Let X be a 2-path of length 2 which originates from a pair of vertices of some A not equal to e^-i or e*. Therefore the smallest structure satisfying these conditions is a 2-path P of length 5 with X starting from the middle clique, A and its length is 3. According to Przulj and Corneil [34], there are three non-isomorphic 2-paths of length 5 which are AgjA^Ag given in Figure 4.12. Since A has just one edge other than e^—i, e*, there is exactly one way to add the first vertex x\ of X. Let x2
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Figure 4.9: Forbidden subgraphs for 2-caterpillar
Figure 4.10: Construction 1
be the second vertex of X. The vertices adjacent to ;r2 are X\ and one of the vertices from N(x i). Thus we get six 2-trees Ai,i = 1,2, 3,...,6 as in Figure 4.13. Observe that Ai, A2, A3, A4 and A5 are isomorphic to B3 and H6 is isomorphic to B2. But G cannot have B‘2,B'3 as induced subgraphs. Hence G cannot have any 2-path of length at least 2 originating from a pair of vertices of some U not equal to e^_ i or e*. Hence from cases (1) and (2) we can conclude that G cannot have any 2-leaf at 2-distance 2 or more. So G must be a 2-caterpillar.
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Figure 4.11: Construction 2
A5-2
A 5-3
Figure 4.12: Construction 3
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Figure 4.13: Six 2-trees called Afs
Theorem 4.3.2 A 2-caterpillar is an interior 2-caterpillar if and only if it does not contain FI as an induced subgraph.
Proof: Let G be a 2-tree which is a 2-caterpillar with [e0,ti,........e„], a longest
2-path of G which we call P. We first assume that it is an interior 2-caterpillar. So it cannot have any 2-leaf adjacent to the end points of a non-interior edge. Since FI is a 2-caterpillar with a longest 2-path being eo, H, F, F, and a 2-leaf originating from a pair of vertices of F not equal to e\ or e-2, it is a non-interior 2-caterpillar. Hence G cannot have FI in it as an induced subgraph.
Now we assume G is a 2-caterpillar without FI. If G is not interior then it has a 2-path of length 1 originating from a pair of vertices of some F ^ Since P
is assumed to be a longest 2-path of G, there exists a 2-path of length at least 1 on
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either side of U on F. Let X be a 2-path of length 1 which originates from a pair of vertices of some U not equal to e^_i or e*. Therefore the smallest structure satisfying these conditions is a 2-path P of length 3 with X starting from the middle clique, f2 and its length is 1. According to Przulj and Corned [34], there is just one 2-path of length 3 which is A\ depicted in Figure 4.10. Thus the vertex x\ that forms X is adjacent to a pair of vertices in i2 such that X\ ^ e1} e2. Thus X\ can be added to P in just one way, as in number 3 of Figure 4.11 which forms FI. This is a contradiction since G does not contain any subgraph isomorphic to FI. Hence G cannot have a 2-leaf at a non-interior edge. So G must be an interior 2-caterpillar.
Theorem 4.3.3 A 2 tree UPIG G is an interior 2-caterpillar
Proof: Let G be 2-tree unit probe interval graph. If G is not a 2-caterpillar then, by theorem 4.3.1, it must containone of Bl, Bl', B2, or B3 as an induced subgraph. Furthermore as seen in Figure 4.9 Bl and BV contains FI, B2 and B3 contains FI as induced subgraphs. Hence G should also contain FI or FI as induced subgraphs provided G is not a 2-caterpillar which is a contradiction since FI and FI are forbidden subgraphs for unit probe interval graphs. Hence G cannot contain Bl, Bl', B2 or B3 as induced subgraphs. So by Theorem 4.3.1 G must be a 2-caterpillar.
We have proved that any 2-caterpillar is interior if and only if it does not contain FI as an induced subgraph. But FI is a forbidden subgraph for unit probe interval graphs. Hence G cannot contain FI as an induced subgraph and so it must be an interior 2-caterpillar. â– 
4.4 Forbidden subgraphs for 2-paths which are unit probe interval graphs
Interior 2-caterpillars can be thought of as 2-paths with 2-leaves on some of the
interior edges. So when characterizing interior 2-caterpillars which are unit probe
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interval graphs the natural step is to characterize 2-paths which are unit probe interval graphs. Once we have a complete characterization of the 2-path unit probe interval graphs we will keep adding 2-leaves at every possible interior edge to determine which structures fail to be a unit probe interval graphs thus deriving a list of forbidden subgraphs for interior 2-caterpillars which are unit probe interval graphs. As we have mentioned earlier the characterization problem for 2-paths in itself turns out to be very difficult. In this section we give a list of forbidden subgraphs for 2-paths which are unit probe interval graphs. The forbidden subgraphs are named as F2, F3, FA, F5, F6, F7, F8, F9, F10 and Fll. All of these subgraphs are probe interval graphs but they fail to have a unit probe interval representation.
It is worth mentioning here that we use either Lemma 4.2.7 or Lemma 4.2.11 for the following proofs. We use Lemma 4.2.11 for the proofs that F2, F3, FA, FA, F5, F6, F10, Fll are forbidden subgraphs for unit probe interval graphs and Lemma 4.2.7 for proving that F7, F8, F9 are also forbidden subgraphs for unit probe interval graphs.
Lemma 4.4.1 The 2-path F2 (5-fan) given in Figure f.lf is not a unit probe interval graph.
Proof: Let us assume that F2 as shown in Figure 4.14 is a UPIG. It can be observed from the figure that U2,uz,u±,u<$,U6,U7 and U2,ui,U3,U4,U5,ue are vertices of two 4-fans and both of them have center at tt2. The central radial vertex of the first 4-fan is u5 and the central radial vertex of the second 4-fan is tt4. Hence by lemma 4.2.11 «4 and u5 must be non-probes. But tt4 u5 and hence we get a contradiction. Thus F2 is not a unit probe interval graph. ■
Lemma 4.4.2 The 2-path F 3 given in Figure f.15 is not a unit probe interval graph.
Proof: Let us assume that F3 is a UPIG. We can see from Figure 4.15 that x_i, x\, xg and x\q are centers of 4-fans and xj and x\\ are central radial vertices
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Figure 4.14: F2
Figure 4.15: F3
of the four 4-fans. Also X-\ and x\ are end radial vertices of the 4-fans with centers at x\ and X-\ respectively. So x\ and X-\ must be probes. Furthermore xg and rrio are also end radial vertices of the 4-fans with centers at rrio and xg respectively. So rrio and xg must be probes as well. Flence X-3, X3, xe and X12 the other end radial vertices of the 4-fans must be non-probes. But ^3 xe and so we get a contradiction.
Lemma 4.4.3 The 2-path F4 given in Figure 4-16 is not a unit probe interval graph.
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X-4 X14
Figure 4.16: F4
Proof: Let us assume that F4 is a UPIG. We can see from Figure 4.16 that ;x_i, X\, x8 and Xu are centers of 4-fans and x_2l x2l Xg and xV2 are central radial vertices of the four 4-fans. Also x_i and X\ are end radial vertices of the 4-fans with centers at X\ and x_i respectively. So they must be probes. Furthermore x8 and Xu are also end radial vertices of the 4-fans with centers at Xu and x8 respectively. So they must be probes too. Flence X-3, x-8, xg and ;ri3 the other end radial vertices of the 4-fans must be non-probes. But ^3 xg and so we get a contradiction. â– 
Lemma 4.4.4 The 2-path F5 given in Figure 4-17 is not a unit probe interval graph.
Proof: Let us assume that F5 is a UPIG. We can see from Figure 4.17 that X-i, xi, x8 and xi2 are centers of 4-fans and X-2l x2l xg and ^13 are central radial vertices of the four 4-fans. Also X-\ and x\ are end radial vertices of the 4-fans with centers at x\ and X-i respectively. So x\ and X-\ must be probes. Furthermore x8 and x\2 are also end radial vertices of the 4-fans with centers at xV2 and x8 respectively. So xV2
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and xg must be probes too. Hence X-3, X3, xg and Xu the other end radial vertices of the 4-fans must be non-probes. But X3 xg and so we get a contradiction. â– 
Lemma 4.4.5 The 2-path F6 given in Figure 4-18 is not a unit probe interval graph.
Proof: Let us assume that F6 is a UPIG. We can see from Figure 4.18 x_2l X\, Xg and xV2 are centers of 4-fans and x_3, x2l x8 and ;ri3 are central radial vertices of the four 4-fans. Hence x_i and Xu cannot be non-probes which are also the end radial vertices for the second and the third 4-fan. So x3 and xe the other end radial vertices of the 4-fans must be non-probes. But x3 xe and so we get a contradiction.
Lemma 4.4.6 The 2-path F7 given in Figure 4-19 is not a unit probe interval graph.
Proof: Let us consider a 2-tree probe interval graph F7 as in Figure 4.19.
Assume that F7 has a unit probe interval representation. Note that U4 is the center
of an induced Ad,3 formed by U4,u2,U3 and ug. So by Lemma 4.2.7 U4 must be a
probe. Note that u.g is also a center of an induced Ad,3 formed by uq, im, U7 and ug.
So Ug must be a probe and at least two of U4, uj and ug must be non-probes. Since
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Figure 4.18: F6
u4 is a probe, uj and ug must be non-probes. But ug is the center of another induced Ad,3 with leaves ue, uw and uu and so ug must be a probe which is a contradiction.
â– 
Lemma 4.4.7 The 2-path F8 given in Figure 4-20 is not a unit probe interval graph.
Proof: Let us consider a 2-tree probe interval graph F8 which is a merge of three 4-fans as in Figure 4.20 with centers at u4,u4i and u6. Let us assume that F8 has a unit probe interval representation. The vertex u4 is the center of an induced Ad,3 formed by u4,u2, u3 and u6 so u4 must be a probe. Note that u6 is also a center of an induced Ad,3 with leaves u4, u7 and u8; so u6 must be a probe and at least two of u4, Uj and ug must be non-probes. Since u4 is already a probe, then u7 and ug must be non-probes. Also 11,9 and u\0 are adjacent to itg, so ug and u4o must be probes. Now, un, ug, uio and U13 form an induced Ad,3 with center un. So at least two vertices out of ug, u4o and U13 must be non-probes. But ug and , u4o are probes, so U13 is the only vertex which is a non-probe and hence we get a contradiction.
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Lemma 4.4.8 The 2-path F9 given in Figure 4-21 is not a unit probe interval graph.
Proof: Let us consider a 2-tree probe interval graph F9 which is a merge of three 4-fans G1,G2 and G3 with centers at X-2,xi and xq as in Figure 4.21. let us assume that F9 has a unit probe interval representation. x_2 is the center of an induced Ad,3 formed by ;r_2, ;r_4, ;r_3 and x0. So ;r_2 must be a probe and either x_3
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Figure 4.21: F9
and rro or X-3 and X-3 are non-probes. Again x\ is the center of an induced Ad,3 formed by x\,X-i,X2 and ^3. So x\ must be a probe and either X-\ and X2 or x-2 and X3 are non-probes.
Case 1: If x_3 and x0 are non-probes, then x_\ must be a probe. So X2 and x3 must be non-probes by Lemma 4.2.7 and 4.2.11 which implies that two adjacent vertices x0 and X2 are both non-probes which is a contradiction. Hence x_3 and x0 cannot be both non-probes.
Case 2: If x_3 and ;r_4 are non-probes then x_\ must be a probe. So X2 and x3 must be non-probes by Lemmas 4.2.7 and 4.2.11 which implies that x4 and x5 are probes. From G3 whose center is at xe we can see that ^6,^4, ^5 and xg form an induced Ad,3 with center xe, and so at least two of X3, X5 or must be non-probes. But x4 and X5 are already probes and so we get a contradiction. Hence It is not possible to construct a unit probe interval representation of F9.
Lemma 4.4.9 The 2-path A10 given in Figure 4-22 is not a unit probe interval graph.
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Figure 4.22: F10
Proof: Let us assume that F10 is a UPIG. We can see from the Figure 4.22 that x_2l X\, x7 and xV2 are centers of 4-fans. It can also be observed that x_3, x2l x8 and Xu are central radial vertices of the four 4-fans, and thus must be non-probes. Hence x_i x0, xg and ;ri0 must be probes. Each of these vertices is also one of the end radial vertices of the 4-fans. So the other end radial vertices ;r_4, x3, xe and ;ri3 of the 4-fans must be non-probes. But x3 xe and so we get a contradiction. â– 
Lemma 4.4.10 The 2-path Fll given in Figure 4-33 is not a unit probe interval graph.
Proof: Let us assume that Fll is a UPIG. We can see from the Figure 4.23
that x_2, xi, x7 and #10 are centers of 4-fans where x_3, x2l x8 and xV2 are central
radial vertices of the four 4-fans. So x_3, x2l x8 and xV2 must be non-probes. Hence
X-i and xg must be probes and they are the end radial vertices of 4-fans with centers
at Xi and x_2. Furthermore x7 and rrio are also end radial vertices of the 4-fans with
centers at rrio and x7 respectively. Since rrio and x7 are also centers, they must be
probes. Hence the other end radial vertices, x_4, ^3, x3 and xn of the 4-fans, must
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X
Figure 4.23: Fll
be non-probes. But ^3 xq and so we get a contradiction. â– 
4.5 2-path unit probe interval graph characterization
In the following section we will give a characterization of 2-paths that are unit probe interval graphs. The 2-path characterization turned out to be quite difficult and, the complete list of forbidden subgraphs for this characterization includes 10 subgraphs. In the beginning of the section we will give several definitions which have been used to proceed through the proofs. We will also introduce some additional definitions as we move towards the final goal. While constructing a 2-path we start with a 2-complete graph, K2 and call it G. Next we keep adding new vertices adjacent to a 2-complete subgraph of G such that the new vertex is adjacent either to the two most recently added vertices or not. If it is adjacent to the most recently added vertices we get a 2-path with no 4-fan which we will call a straight 2-path ( we will formally define a straight 2-path later). If not, then the new vertex, after addition, either forms an FI which is not allowed since it is not a 2-path or it creates a 4-fan. So at every stage we have exactly three places where the new vertex can be added provided
no induced F2.....Fll is formed and as a result we either get straight 2-paths, FIs
or merge of 4-fans. Hence it can be concluded from the discussion so far that with
addition of a vertex to an existing 2-path either a 4-fan is created or not. Thus the
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structures of a 2-path avoiding the forbidden subgraphs from F2,.....Fll, can either
be a straight 2-path or merge of 4-fans. At first we figure out the possible ways 4-fans can be merged to each other.
Definition 4.5.1 Edge-consecutive 4-fans: Let Ai}Ai+1 be two 4-fans. The 4-fans Ai,Ai+1 will be called edge-consecutive if Ai shares at least one edge with Ai+1, i=l,....(n-l).
Definition 4.5.2 Vertex-consecutive 4-fans: Let Ai}Ai+1 be two 4-fans. The 4-fans Ai,Ai+1 will be called vertex-consecutive if Ai shares exactly one vertex with Ai+1.
Now we will prove some useful lemmas using the above definitions to find the possible shapes of a 2-path. First we will deal with 2-paths which are just merge of 4-fans. As stated earlier 4-fans in a 2-path can be either edge-consecutive or vertex-consecutive. First we determine the possible structures of edge-consecutive 4-fans.
Lemma 4.5.3 Two edge-consecutive 4-fans without an F2 (5-fan) cannot share more than two K3s.
Proof: Let t\ — t2 — t3 — hi (eo,ti,.....f4,e4) be a 4-fan A. let us add another
4-fan B to this such that t2, t3 and f4 also belongs to B. In other words t2, t3 and hi are the shared K3. The only way we get a 4-fan from f2> £3 and £4 is by making a new vertex adjacent to e4 as shown in picture (1) of the Figure 4.25. It is easy to see that we get an F2 here which is a contradiction. Hence two 4-fans cannot share more than two K3 s. ■
Lemma 4.5.4 Two edge-consecutive 4-fans without an F2 (5-fan) can share two K3s in exactly one way as A\ in the Figure 4-%4 •
Proof: Let Al and A2 be two edge-consecutive 4-fans without an F2 (5-fan)
such that they share exactly two K3s. Let us call this graph G. Since Al and A2
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Figure 4.24: Non-isomorphic 2-paths of length 6
share exactly 2 K3s, then G must contain exactly six K3s. In other words G must be a 2-path of length 6. It has been proved in Przulj and Corneil that there are six non-isomorphic 2-paths of length 6 (Hg, H|, Hj, A%, H|, H|) as shown in Figure 4.24. It can be seen in the figure that A\ contains no 4-fan, and that H|, A% have exactly one induced 4-fan. Further Hj,Hg have an induced F2 in them. Thus A\, H|, H|, Hj, cannot be G. Hence G must be A$ â– 
The following lemma will determine the possible structures any two edge-consecutive 4-fans without an F2 can have.
Lemma 4.5.5 If G1 and G2 are two edge-consecutive 4-fans that form a 2-path with no F2, then G1 and G2 together are one of the graphs in Figure 4-27.
Proof: Let t\ — t2 — t3 — t3 (eo,H,....TiTi) be a 4-fan. Call it G\. Let G2
be another 4-fan. We can make these two 4-fans edge-consecutive without forming an induced F2 in the following ways ( keeping in mind that they cannot share more than two 3-cliques):
Casel: G1 and G2 share exactly two K3s:
This case has been dealt with in the previous lemma. Its construction is illustrated in the pictures 2(a), 2(b), 2(c) of Figure 4.25. 2(b) and 2(c) contain an FI ( which is not a 2-path)and an F2 and so we do not consider them.
Case2: G1 and G2 share exactly one K3:
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4
tl-t2-t3-t4
tl'-t4-t3-t2
tl-t2-t3-t4
tl'-t2'-t3-t4
tl-t2-t3-t4 tl-t2-t3-t4
tl'-t2'-t3-t4 tl'-t2'-t4-t3
tl-t2-t3-t4
tl,-t2,-t3,-t4
tl-t2-t3-t4
tl,-t2,-t3,-t41
tl-t2-t3-t4
tl,-t2,-t3,-t41
tl-t2-t3-t4
tl,-t2'-t3,-t4'
tl-t2-t3-t4
tl,-t2l-t3,-t41
Figure 4.25:
Construction that shows possible merge of two-4-fans
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In this case we will merge G\=t\ — f2 — G — G with G2 such that the 3-clique G becomes the first 3-clique of G2. Since G has two available edges to which the vertex from G2 can be adjacent, we get exactly 2 ways to add it. One of the ways form an F2 as shown in picture 3(c) of Figure 4.25 and so we do not consider it any more. Thus we construct G — t'3 of G2 where t'3 is the second 3-clique of G2. Now at this stage we are just left with 2 options for the next vertex from G2 which again in turn has just one option in each case for the final vertex of G2. Thus we get the structures 3(a) and 3(b) of the Figure 4.25. 3(b) also has an induced F2 and so we no longer consider it.
Case3: Gi and G2 share exactly one edge:
In this case we will merge G\=t\ —12 — G — G with G2 such that the edge e4 becomes the first edge of G2. At this stage we have exactly one way to add a new vertex. After this we have exactly two ways to add the new vertex. One leads to 4(a), 4(b) and the other leads to 4(c), 4 (d) of Figure 4.25. This construction is illustrated in Figure 4.26. Here 4(b) and 4(d) have an induced F2 and so we do not consider them. 4(c) is an extension of 2(a).
So our final structures are given by 2(a), 3(a) and 4(a) as illustrated in Figure 4.25.
â– 
Note: 2(a) of Figure 4.25 is formed if two edge-consecutive 4-fans share exactly two K3l 3(a) of Figure 4.25 is formed if two edge-consecutive 4-fans share exactly one K3 and 4(a) of Figure 4.25 is formed if two edge-consecutive 4-fans share exactly one edge.
Lemma 4.5.6 If a graph is a merge of three graphs Gi, G2, G3 which are three edge-consecutive f-fans that form a 2-path with no F2, F7, F8, F9 then it is one of the following graphs in Figure f.31.
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Proof: We consider 3 cases here:
Casel: G\ and G2 share exactly two K3s and G3 shares either two Kas, one Ka or one edge with G2. Since 2(a), 3(a) and 4(a) as illustrated in Figure 4.25 are the possible structures here. We start with G1G2 as the graph 2(a) in Figure 4.25. So now we have to look at the case where G2 can be edge consecutive with G3 with no F2. We know (by Lemma 4.5.5) that it can be done in exactly 3 ways and hence we get 3 structures as shown in Figure 4.28.
Case2: G1 and G2 share exactly one K3 and G3 shares either two K3s, one K3 or one edge with G2. Since 2(a), 3(a) and 4(a) as illustrated in Figure 4.25 are the possible structures here. We start with G3G2 as the graph 3a in figure 4.25. So now we have to look at the case where G2 can be edge consecutive with G3 with no F2. We know (by lemma 4.5.5) that it can be done in exactly 3 ways and hence we get 3 structures as shown in the Figure 4.29.
Case3: Gi and G2 share exactly one edge and G3 shares either two K3s, one K3 or one edge with G2. Since 2(a), 3(a) and 4(a) as illustrated in Figure 4.25 are the possible structures here. We start with G1G2 as the graph 4(a) in figure 4.25. As in the previous cases, now we look at the case that G2 can be edge consecutive with G3 with no F2. Again we know (by Lemma 4.5.5) that it can be done in exactly 3 ways and hence we get 3 structures as shown in Figure 4.30.
Since (3),(2) from Figure 4.28 are isomorphic to (7) from Figure 4.30, and (4) from Figure 4.29 respectively and (6) from Figure 4.29 is isomorphic to (8) from figure 4.30, we get six non-isomorphic structures which are illustrated in figure 4.31 three of which are forbidden subgraphs F7, F8, F9.
â– 
We now look at the vertex-consecutive case. We already know that edge-consecutive 4-fans are capable of assuming exactly 3 structures. The following two lemmas will prove that vertex-consecutive 4-fans can assume exactly one structure
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Figure 4.26: Construction of two edge-consecutive 4-fans
Figure 4.27:
All possible two edge-consecutive 4-fans and their representations
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2K3 and S
2K3 and 1 ed9e
Figure 4.28: Formation of three edge-consecutive-4-fans
Figure 4.29: Formation of three edge-consecutive-4-fans
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Figure 4.30: Formation of three edge-consecutive-4-fans
six non-isomorphic possible sturctures of three edge consecutive 4-fans
9
Figure 4.31: three edge-consecutive-4-fans
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which is also illustrated below.
Lemma 4.5.7 If G1 and G2 are two vertex-consecutive 4-fans that form a 2-path with no F2 then it is the graph in Figure 4-33.
Proof: Let G be the merge of Gi and G2. Since G forms a 2-path, it cannot have FI as an induced subgraph. It can be observed that G1 and G2 cannot share any non-end radial vertex since in that case three fd3s from one and one fd3 from its complement would form an FI. Hence G1 and G2 can share only the end vertices. Hence if G1= U\ — 112 — u8 — W4 — W5 — ua with center at u4 and G2= Uq — Uj — u8 — Ug — uw — Un with center at u8 then xx6 is the shared vertex. Thus we get the possible structure of G which is illustrated in Figure 4.33.
A construction of G from G1 is also illustrated in figure 4.32. â– 
Lemma 4.5.8 If Gl, G2, G3 are vertex-consecutive 4-fans that form a 2-path vnth no F2 then it is the graph in Figure 4-34-
Proof: G\ and G2 share exactly one vertex and G3 shares one vertex with G2. So now we have to look at the ways G2 can be vertex-consecutive with G3 with no F2. From the previous lemma we know that it can be done in exactly 1 way and hence we get only one structure as shown in Figure 4.34.
â– 
Now we define some possible structures of a 2-path which have nice unit probe interval representations. The representations are also given below. We will also use the operation + to denote merge between 4-fans. The sign plus is used mostly in Figures for convenience.
Definition 4.5.9 Straight 2-path : A straight 2-path is a 2-path vnth no 4-fans in the structure. An example is illustrated in Figure 4-37.
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u3
u5
This is isomorphic to 3 edge consecutive 4-fans
The only structure possible
Figure 4.32: Formation of two vertex-consecutive-4-fans
Figure 4.33: The only structure possible for two vertex-consecutive-4-fans
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Figure 4.34: Representations of three vertex-consecutive-4-fans
Figure 4.35: Another representation of three vertex-consecutive-4-fans
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Figure 4.36: Three possible structures of two edge-consecutive 4-fans
Definition 4.5.10 Staircase : A 2-path is a staircase if it is a merge of 4-fans where the consecutive 4-fans share exactly two 3-cliques. An example is illustrated in Figure 4-36a.
Definition 4.5.11 2-snake : A 2-path is a 2-snake if it is a merge of 4-fans where the consecutive 4-fans share exactly two or one 3-cliques and the centers of all the 4-fans form a path. An example is illustrated in Figure 4-36b.
Definition 4.5.12 Extended staircase :A 2-path is an extended staircase if it is a merge of 4-fans where the consecutive 4-fans share exactly one edge. This is illustrated in Figure 4 -36c.
The following lemma is a very useful result and helps a lot in later proofs where we give the unit probe interval representations of the 2-paths.
Lemma 4.5.13 The 2-leaves on both ends of a 2-snake and an extended staircase must always be non-probes.
Proof: Let us first consider a 2-snake as shown in Figure 4.27 (1). Here u4 and Uq are the centers of two 4-fans, so they must be probes. Again u4 and u6 are also one of the end radial vertices of the 4-fans with center u6 and u4. Hence the other end radial vertices of the 4-fans u-2 and ug which are also the end 2-leaves of the 2-snake must be non-probes.
Let us now consider an extended staircase as shown in Figure 4.27 (2). Here X-2
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Figure 4.37: Straight 2-path
and x\ are the centers of two 4-fans. So they must be probes. The central radial vertices X-3 and X2 of the two 4-fans must be non-probes. So their adjacent vertices X-i and rro must be probes. But X-\ and rro are also one of the end radial vertices of the 4-fans with centers at x\ and X-2 respectively. Hence the other end radial vertices , ;r_4 and x3, which are also the end 2-leaves of the extended staircase must be non-probes. â– 
Lemma 4.5.14 A straight 2-path is a unit probe interval graph.
Proof: It can be easily seen that a straight 2-path does not have any induced Ad,3. Hence it is a unit interval graph. Therefore it is a unit probe interval graph. â– 
Lemma 4.5.15 A staircase is a unit probe interval graph.
Proof:
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staircase+staircase
Xi
x2
X3
V
X5
X6
X7
X8
X11
X12
Figure 4.38: Staircase
2-snake+2-snake
Figure 4.39: 2-snake
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extended staircase+ staircase+extended staircase+staircase
extended staircase+staircase
Figure 4.40: Extended-staircase+staircase
Xi
Figure 4.41: Staircase representation


Figure 4.42: 2-snake representation
X3 X5
3 : extended staircase+ staircase
Figure 4.43: extended-staircase+staircase representation
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Full Text

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ONCHARACTERIZATIONSANDSTRUCTUREOFINTERVALDIGRAPHS ANDUNITPROBEINTERVALGRAPHS by ShilpaDasgupta M.S.,UniversityofColoradoDenver,2007 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy AppliedMathematics 2012

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ThisthesisfortheDoctorofPhilosophydegreeby ShilpaDasgupta hasbeenapproved by J.RichardLundgren,AdvisorandChair MichaelS.Jacobson MichaelFerrara WilliamE.Cherowitzo DavidEBrown Date ii

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Dasgupta,ShilpaPh.D.,AppliedMathematics OnCharacterizationsandStructureofIntervalDigraphsandUnitProbeInterval Graphs ThesisdirectedbyProfessorJ.RichardLundgren ABSTRACT Intervalgraphsandtheirvariationshavebeenstudiedextensivelyforthelast50 yearsfrombothatheoreticalstandpointandduetotheirimportanceinapplications. Inthisthesiswewillexploreseveralvariationsofintervalgraphsandunitinterval graphsincludingintervaldigraphs,intervalbigraphsandprobeintervalgraphs.IntervalgraphsweremathematicallyintroducedbyHajos[22].Theoriginofinterval graphsmayalsoberegardedasanapplicationtotheresearchofBenzerin1959[1] inhisstudyofthestructureofbacterialgenes.Manynicecharacterizationshave beenfoundforintervalgraphs,butthesamecannotbesaidaboutthevariousgeneralizations.Theworkpresentedinthisthesiswillindicatethatcharacterizationof thebroadestclassesofthegeneralizationsposesaverydicultproblemandsothe focusinthisthesisisonsubclassesofintervaldigraphs,intervalbigraphs,andprobe intervalgraphs. IntervaldigraphswereintroducedbySenetal.in[39].In1998Westgave adjacencymatrixcharacterizationsofintervaldigraphsandunitintervaldigraphs [42].Sofar,inthemostgenericsense,noforbiddensubgraphcharacterizationof intervaldigraphshasbeenbeenfound,butthosetournamentswhichareintervaldigraphshavebeencharacterizedinvariouswaysbyBrown,BuschandLundgrenin 2007.Inthisthesiswegeneralizesomeoftheirresultstootherclassesofinterval digraphs. Anaturalextensionofintervalgraphs,calledintervalbigraphs,wereintroduced byHarary,Kabell,andMcMorris[23]in1982.Unitandproperintervalgraphswere iii

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introducedandcharacterizedbyRobertsin1969[36].Heprovedthattheclassesof properintervalgraphsandunitintervalgraphscoincide.ProskurowskiandTelle[33] extendedtheideaofproperintervalgraphstop-properintervalgraphs.Recently BeyerlandJamisonintroducedp-improperintervalgraphs.Theyfocussedonaspecialcaseofp-improperintervalgraphs.Herewegeneralizeandobtainsimilarresults forp-improperintervalbigraphs. Theprobeintervalgraphwasinventedinordertoaidwiththetaskcalledphysicalmappingfacedinconnectionwiththehumangenomeproject,cf.workofZhang andZhangetal.[43],[45],[44].Nogeneralizedcharacterizationofprobeinterval graphshavebeenfoundsofar.LiShengrstcharacterizedprobeintervalgraphsfor trees[41].Followingthischaracterizationofcycle-freeprobeintervalgraphs,in2009 Brown,ShengandLundgrengaveacharacterizationforcycle-freeunitprobeinterval graphs[15].RecentlyBrownandLangleycharacterizedunitprobeintervalgraphs forbipartitegraphs[10].Inthisthesiswegiveacharacterizationof2-treesthatare unitprobeintervalgraphs.In2005PrzuljandCorneilfoundatleast62distinct minimalforbiddeninducedsubgraphsforprobeintervalgraphsthatare2-trees[34]. MorerecentlyBrown,FleschandLundgrenextendedthelistto69andalsogavea characterization[7].InthisthesiswefollowtheideaofPrzuljandCorneilforthenext bestcasewhichisfor2-trees.Therestrictionto2-treesisareasonablethingtodo becauseoftheprogressionoftheSheng,Brown-Sheng-Lundgren,andBrown-Langely results,andalsobecauseoftheCorneil-PrzuljandBrown-Flesch-Lundgrenresults. Werstcharacterize2-caterpillarsandinterior2-caterpillarsintermsofforbidden inducedsubgraphsandshowthat2-treesthatareunitprobeintervalgraphshaveto beinterior2-caterpillars.Thenwelookat2-pathsthatareunitprobeintervalgraphs andcharacterizethem.Usingsimilarideaswesubsequentlycharacterizeinterior2caterpillarwhichareunitprobeintervalgraphs.Finallyweusetheseresultstogeta completecharacterizationof2-treeswhichareunitprobeintervalgraphs. iv

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Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:J.RichardLundgren v

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DEDICATION MomanddadandmysweetheartsIsha,Sheila,AnoushkaandBubai. vi

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ACKNOWLEDGMENTS FirstofallIwouldliketothankmyadviser,Dr.J.RichardLundgren,forguiding, mentoringandengagingmeinmathematics.Iwouldalsoliketothankmyprofessors, friendsandcolleagueswithwhomI'veworkedoverthepastfewyears.Lastly,and certainlymostimportantly,Ithankmyfamily,especiallymysisterforherloveand support. vii

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TABLEOFCONTENTS Figures.......................................ix 1.Introduction...................................1 1.1Background..................................1 1.2Introductiontotheresearch.........................5 2.IntervalDigraphs................................7 2.1Introduction..................................7 2.2MotivationfromIntervalTournaments...................11 2.3DigraphswithaTransitive n )]TJ/F15 11.9552 Tf 11.955 0 Td [(2-subtournament............14 2.4Morethanonenontrivialstrongcomponent................22 3.IntervalBigraphImpropriety..........................27 3.1Introduction..................................27 3.2ImproprietyandWeightofIntervalBigraphs................29 3.3 p -criticalintervalbigraphs..........................36 3.4Conclusionsandfuturework.........................47 4.Characterizationofunitprobeinterval2-trees................48 4.1Introduction..................................48 4.2Preliminaries.................................49 4.3Characterizationof2-caterpillars......................56 4.4Forbiddensubgraphsfor2-pathswhichareunitprobeintervalgraphs..61 4.52-pathunitprobeintervalgraphcharacterization.............70 4.6forbiddensubgraphsforinterior2-caterpillarswhichareunitprobeinterval graphs..................................119 4.7Characterizationofinterior2-caterpillarswhichareunitprobeinterval graphs..................................130 References ......................................172 viii

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FIGURES Figure 2.1Intervaldigraph................................9 2.2Zeropartition.................................10 2.3Intervalbigraph................................10 2.4Insect.....................................11 2.5Bigraphrepresentationofadigraph.....................11 2.6 D )]TJ/F19 11.9552 Tf 11.956 0 Td [(v .....................................13 2.7ATEinthebigraph..............................15 2.86-cycle.....................................16 2.910-cycle....................................16 2.10Induced6-cycleinthebigraphrepresentation...............17 2.11Morerestrictionsonthearcs........................17 2.12Exampleofadigraphsatisfyingaconditionofthetheorem2.3.1andits zeropartition................................25 2.13Exampleofadigraphsatisfyingaconditionofthetheorem2.3.2andits zeropartition.................................26 2.14Exampleofadigraphthatfailsthehypothesisoftheorem2.3.2since v 0 beats v 00 andhenceformsa6-cycleinitsbigraphrepresentation.....26 2.15Strongcomponents..............................26 3.1 K 1 ; 3 ......................................29 3.2 p -ImproperIntervalGraph.........................30 3.30-ImproperIntervalBigraph........................30 3.4ForbiddensubgraphsforproperIntervalbigraphwiththeirintervalrepresentationstoforcecontainmentfrom X .................31 3.5IntervalrepresentationsofB1,B2,B3toforcecontainmentfrom Y ...31 3.6Improprietyof imp B 1 =2, imp B 2 =1................33 ix

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3.7Exteriorlocalcomponents.........................33 3.8Intervalbigraphwithimpropriety1....................34 3.9Intervalbigraphwithimpropriety2....................34 3.10Calculationofweight............................35 3.11Intervalbigraphwithweight2.......................35 3.12Changesinweightduetovariationindenition.............37 3.13Intervalbigraphwhere wt X =0
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4.15F3.......................................63 4.16F4.......................................64 4.17F5.......................................65 4.18F6.......................................66 4.19F7.......................................67 4.20F8.......................................67 4.21F9.......................................68 4.22F10......................................69 4.23F11......................................70 4.24Non-isomorphic2-pathsoflength6.....................72 4.25Constructionthatshowspossiblemergeoftwo-4-fans...........73 4.26Constructionoftwoedge-consecutive4-fans................76 4.27Allpossibletwoedge-consecutive4-fansandtheirrepresentations....76 4.28Formationofthreeedge-consecutive-4-fans.................77 4.29Formationofthreeedge-consecutive-4-fans.................77 4.30Formationofthreeedge-consecutive-4-fans.................78 4.31threeedge-consecutive-4-fans........................78 4.32Formationoftwovertex-consecutive-4-fans.................80 4.33Theonlystructurepossiblefortwovertex-consecutive-4-fans.......80 4.34Representationsofthreevertex-consecutive-4-fans.............81 4.35Anotherrepresentationofthreevertex-consecutive-4-fans.........81 4.36Threepossiblestructuresoftwoedge-consecutive4-fans.........82 4.37Straight2-path................................83 4.38Staircase....................................84 4.392-snake....................................84 4.40Extended-staircase+staircase........................85 4.41Staircaserepresentation...........................85 xi

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4.422-snakerepresentation............................86 4.43extended-staircase+staircaserepresentation................86 4.44Fouredge-consecutive4-fans,structure1..................91 4.45Fouredge-consecutive4-fans,structure2..................92 4.46Fouredge-consecutive4-fans,structure3..................92 4.47Exampleofmergesfordierentvaluesof n ................97 4.48 n =1:edge-consecutivemergefor2-paths.................97 4.49 n =1:edge-consecutivemergefor2-paths.................98 4.50 n =1:edge-consecutivemergefor2-paths.................99 4.51 n =2:edge-consecutivemergefor2-paths.................100 4.52 n =2:edge-consecutivemergefor2-paths.................101 4.53 n =2:edge-consecutivemergefor2-paths.................101 4.54 n =2:edge-consecutivemergefor2-paths.................102 4.55 n =2:edge-consecutivemergefor2-paths.................103 4.56 n =3:edge-consecutivemergefor2-paths................103 4.57 n =3:edge-consecutivemergefor2-paths................104 4.58 n =3:edge-consecutivemergefor2-paths.................104 4.59 n =0:vertex-consecutivemergefor2-paths................108 4.60 n =1:vertex-consecutivemergefor2-paths................109 4.61 n =2:vertex-consecutivemergefor2-paths................110 4.62 n =3:vertex-consecutivemergefor2-paths...............110 4.63vertex-consecutive-edge-consecutivefor2-path...............112 4.64vertex-consecutive-edge-consecutivefor2-path...............113 4.65 n =1:vertex-consecutive-edge-consecutivefor2-path..........113 4.66 n =2:vertex-consecutive-edge-consecutivefor2-path..........114 4.67 n =2:vertex-consecutive-snake-edge-consecutivefor2-path.....114 4.68Straight-2-path+end-2-leaves........................116 xii

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4.69Staircase+end-2-leaves...........................117 4.702-snake+end-2-leaves.............................118 4.71Non-proberestrictionin4-fans.......................120 4.72Non-proberestriction............................121 4.73Unitprobeintervalrepresentation.....................122 4.743-fanwitha2-leaf..............................122 4.75E2.......................................123 4.76probeintervalrepresentationforE2.....................123 4.77E3.......................................124 4.78E4.......................................125 4.79E5.......................................126 4.80ProbeintervalrepresentationofE5.....................126 4.81E6.......................................127 4.82ProbeintervalrepresentationofE6.....................127 4.83E7.......................................128 4.84E8.......................................129 4.85E9.......................................129 4.86E10......................................130 4.87Group-E11-u1,u2areend2-leavesof B i and B j .............131 4.88Possibleplacesof2-leaves..........................132 4.89Representationof2-snakewithallpossible2-leaves............133 4.90Representationofextendedstaircasewithall2-leaves..........133 4.91Representationofstaircasewithall2-leaves................134 4.92Representationofvertex-consecutive4-fanswithall2-leaves.......134 4.93Representationof2-snakewithall2-leaves.................135 4.94Representationofextended-staircaseandstaircasewithall2-leaves...135 4.95Edge-consecutive4-fanswithall2-leaves..................136 xiii

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4.964-fanwithall2-leaves............................136 4.97Vertexconsecutive4-fanswithall2-leaves.................142 4.98 n =1: I 1 ...................................150 4.99 n =1: I 1 ...................................151 4.100 n =2: I 1 ...................................152 4.101 n =3: I 1 ...................................153 4.102 n =4: I 1 ...................................154 4.103Representationofstraight2-pathwithall2-leaves............154 4.104 n =0: I 2 ...................................159 4.105 n =1: I 2 ...................................160 4.106 n =2: I 2 ...................................160 4.107 n =3: I 2 ...................................161 4.108 n =0: I 3 ...................................165 4.109 n =1: I 3 ...................................166 4.110 n =2: I 3 ...................................166 4.111Interior2-caterpillarswithfewer2-leaves.................167 xiv

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1.Introduction 1.1Background Intervalgraphsandtheirvariationshavebeenstudiedextensivelyforthelast 50yearsfromatheoreticalstandpointandduetotheirimportanceinapplications. Inthisthesiswewillexploreseveralvariationsofintervalgraphsnamelyinterval digraphs,intervalbigraphsandprobeintervalgraphs.Theworkonintervalbigraphs willfocuson p -improperintervalbigraphandtheworkonprobeintervalgraphswill beonunitprobeintervalgraphs. Letagraph G havevertexset V G andedgeset E G .If x;y 2 V G are adjacent,thenwedenote xy 2 E G .If G isbipartite,wedenotethepartitionsof thevertexsetas V G = f X [ Y g .Agraphis interval iftoeveryvertex, v 2 V G , wecanassignanintervaloftherealline, I v ,suchthat xy 2 E G ifandonlyif I x I y 6 = ; .Intervalgraphsweretheoreticallyintroducedin1957byHajos[22]and alsoappearedinappliedresearchbyBenzerin1959[1]inhisstudyofthestructureof bacterialgenes.Intervalgraphswerecharacterizedbytheabsenceofinducedcycles largerthan3andasteriodaltriplesbyLekkerkerkerandBoland[26]in1962.An asteroidaltriple ATin G isaset A ofthreeverticessuchthatbetweenanytwo verticesin A thereisapathin G thatavoidsallneighborsofthethird. OtherusefulcharacterizationsofintervalgraphsweregivenbyGilmoreandHomanin1964[19]andFulkersonandGross[17].Extensivestudyandresearchhas beendoneonintervalgraphsforseveraldecadesbybothmathematiciansandcomputerscientists.Thesegraphsareusedtoprovidenumerousmodelsindiverseareas suchasgenetics,psychology,sociology,archaeology,orscheduling.Formoredetails onintervalgraphsandtheirapplications,seebooksbyRoberts[36],Golumbic[20] andMckeeandMcMorris[29]. IntervaldigraphswereintroducedbySenetal.in[39].Langley,Lundgren, MerzgaveresultsonCompetitionGraphsofIntervalDigraphsin[25].They 1

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showedthatthecompetitiongraphofanintervaldigraphisanintervalgraphandthat everyintervalgraphisinfactthecompetitiongraphofsomeintervaldigraph.Lin, SenandWestgavesomeinterestingresultsonIntervaldigraphsand,1-matrices in[27].In1998Westgaveadjacencymatrixcharacterizationsofintervaldigraphs andunitintervaldigraphs[42].Atthispointthereisnoforbiddensubgraphcharacterizationofintervaldigraphs,butrecently,in2007,intervaltournamentswere characterizedbyacompletelistofforbiddensubtournamentsbyBrown,Busch,and Lundgren[6].Inchapter2wegeneralizesomeoftheresultsfromthispaper. Anaturalextensionofintervalgraphs,calledintervalbigraphs,wereintroduced byHarary,Kabell,andMcMorris[23]in1982.Let G beabipartitegraphwithbipartition X [ Y ;wemaywrite G = X;Y;E todenotethis.Abipartitegraph G is anintervalbigraphiftoeveryvertex, v 2 V G ,wecanassignanintervalofthereal line, I v ,suchthat xy 2 E G ifandonlyif I x I y 6 = ; and x 2 X and y 2 Y .Interval bigraphshavebeenstudiedbyseveralauthors[9],[14],[23],[24],[27],and[31]. Initiallyitwasthoughtthatthenaturalextensionofasteriodaltriplesofverticesto asteriodaltriplesofedgesalongwithinducedcycleslargerthan4wouldgiveaforbiddensubgraphcharacterization[23].However,Muller[31]foundinsectsandHell andHuang[24]foundedgeasteriodsandbugsasforbiddensubgraphs,andtodatea completecharacterizationremainselusive.Threeedges a , c and e ofagraph G form an asteriodaltripleofedges ATEifforanytwothereisapathfromthevertexset ofonetothevertexsetoftheotherthatavoidstheneighborhoodofthethirdedge. CyclefreeintervalbigraphswerecharacterizedbyBrownetalin2001[14],andATEs playedasignicantroleinthatcharacterization.In2002ageneralizationofinterval bigraphcalledaninterval k -graphwasintroducedbyBrownetal[8].Morerecently, in2010,LundgrenandTonnsencharacterized2-treesthatareinterval k -graphs[28]. 2

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Agraphisaprobeintervalgraphifthereisapartitionof V G intosets P and N andacollection f I v : v 2 V G g ofintervalsof R suchthat,for u;v 2 V G , uv 2 E G ifandonlyif I u I v 6 = andatleastoneof u or v belongsto P .Thesets P and N arecalledtheprobesandnonprobes,respectively.Anintervalgraphisa probeintervalgraphwithanemptynon-probesetandthisclassofgraphshasbeen studiedextensively.Theprobeintervalgraphmodelwasinventedinordertoaid withthetaskcalledphysicalmappingfacedinconnectionwiththehumangenome projectcf.workofZhangandZhangetal.[43],[45],[44].ThepaperbyMcMorris,WangandZhang[30]hasresultssimilartothoseforintervalgraphsfound in[17]byFulkersonandGrossand[20]byGolumbic.Forexampleintervalgraphs arechordalwhileprobeintervalgraphsareweaklychordalandmaximalcliquesare consecutivelyorderableinintervalgraphswhilequasi-maximalcliquesworksinprobe intervalgraphs.In1999LiShengrstcharacterizedcycle-freeprobeintervalgraphs [41]Theclassesofgraphsrelatedtoprobeintervalgraphsarediscussedin[11]by BrownandLundgren,[8]byBrown,FlinkandLundgren,and[21]byGolumbicand Lipshteyn.Relationshipsbetweenbipartiteprobeintervalgraphs,intervalbigraphs andthecomplementsofcirculararcgraphsarepresentedin[11].Theproblemof characterizinggenericprobeintervalgraphsintermsofforbiddensubgraphsfornow appearstobedicult.Sincetreesweretheonlyclassofgraphswhereprobeinterval graphswerecharacterized,anaturalnextstepwastolookataclassof2-trees.In 2005PrzuljandCorneilattemptedaforbiddensubgraphcharacterizationof2-trees thatareprobeintervalgraphsandtheyfoundatleast62distinctminimalforbiddeninducedsubgraphsforprobeintervalgraphsthatare2-trees[34].Morerecently Brown,FleschandLundgrenextendedthelistto69andgaveacharacterizationin termsofsparsespinyinterior2-lobsters[7]. Inthelast40yearsasubclassofintervalgraphshasbeeninvestigatedandstudied 3

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extensively.ThisclassistheclassofunitintervalgraphsintroducedbyRobertsin 1969.Aunitintervalgraphisanintervalgraphwithallintervalsinsomeinterval representationhavingthesamelength.Aproperintervalgraphisanintervalgraph forwhichthereisanintervalrepresentationwithnointervalproperlycontaininganother.Roberts[36]provedthattheclassesofproperintervalgraphsandunitinterval graphscoincideandheshowedthatintervalgraphsthatcontainno K 1 ; 3 areunitintervalgraphs.In1999BogartandWestgaveashorterproofofthesameresultofthe equalityofproperandunitforintervalgraphs[4].Theygaveaconstructiveproofof thisresult,whereproperintervalsaregraduallyconvertedintounitintervals.Much morerecentlyin2007GardigaveamuchshorternewconstructiveproofofRoberts originalcharacterizationofunitintervalgraphs[18]. Aunitintervalbigraphisanintervalbigraphwithallintervalsinsomeinterval representationhavingthesamelengthandaproperintervalbigraphisanintervalbigraphforwhichthereisanintervalrepresentationwithnointervalcontaininganother properly.Severalcharacterizationsofproperintervalbigraphshavebeenfoundinthe lastdecadeincludingonebyLinandWest[27].Theideaofproperintervalgraphs wasnaturallyextendedto p -properintervalgraphsbyProskurowskiandTelle[33]. The p -properintervalgraphsaregraphswhichhaveanintervalrepresentationwhere nointervalisproperlycontainedinmorethan p otherintervals.Any0-improper intervalgraphisaproperintervalgraphanditiseasytocheckthat K 1 ; 3 isaisa 1-improperintervalgraph.Inchapter3wewillgeneralizetheclassof p -improper intervalgraphto p -improperintervalbigraph.Unitintervaldigraphswerecharacterizedin1997byLinetal[27].In2002,Brownetal[9]conjecturedacharacterization ofunitintervalbigraphs.ThisconjecturewasprovedbyHellandHuang[24]in2004. Morerecentlyin2011BrownandLundgrengaveseveraladditionalcharacterization ofunitintervalbigraphs[12].CurrentlyBrown,FleschandLundgrenareworkingon unitinterval k -graphsfor2-trees.FollowingupSheng'scharacterizationofcycle-free 4

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probeintervalgraphs,in2009,Brown,LundgrenandShenggaveacharacterization ofcycle-freeunitprobeintervalgraphs[15].Twonaturalextensionsarise,therstis tocharacterizebipartitegraphsthatareunitprobeintervalgraphsandsecondlyto characterize2-treesthatareunitprobeintervalgraphs.RecentlyBrownandLangley characterizedunitprobeintervalgraphsforbipartitegraphs[10].Inchapter4we giveacharacterizationof2-treesthatareunitprobeintervalgraphs. 1.2Introductiontotheresearch Wewillintroducenotationaswemovethroughthethesis.Attheinitialstagewe willuseconventionaldenitions.Asmentionedearlierthestudyofintervalgraphs haslongbeenamuchresearchedareainGraphTheory.Themajorityofworkinthis thesisisaneorttocharacterizevariousclassesofvariationsofintervalgraphsand unitintervalgraphs.Inchapter2westudydierentclassesofdigraphsanddeterminethescenarioswhentheyareboundtobeinterval.Sincetheproblemofnding acompletecharacterizationofintervaldigraphsisextremelydicult,weconcentrate onspecicclassesofintervaldigraphs.WefocusourresearchinthischapteronapaperofBrown,Busch,andLundgren[6]whereacompletecharacterizationofinterval tournamentsisfound.Welookatorientedgraphswithcertainpropertiesandnd restrictionsthatneedtobeimposedonitsstructureforittobeinterval.Wendseveralclassesofdirectedgraphson n verticeswhichareintervalwithrestrictionssuch ascontainingtransitivesub-tournamentson n )]TJ/F15 11.9552 Tf 12.26 0 Td [(2or n )]TJ/F15 11.9552 Tf 12.26 0 Td [(1verticesandspecic adjacenciesbetweenvertices. Asstatedearlier,anothernaturalextensionofintervalgraphs,calledintervalbigraphs,wereintroducedbyHarary,Kabell,andMcMorrisin1982.The p -improper intervalgraphs,wherenointervalcontainsmorethan p otherintervals,wereinvestigatedbyBeyerlandJamison[3].Inchapter3weextendtheideabyintroducing p -improperintervalbigraphs,wherenointervalcontainsmorethan p otherintervals ofthesamepartiteset.Inthischapterwedeterminerestrictionsonthestructureof 5

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anintervalbigraphforittobea p -improperintervalbigraph.Wealsostudyspecial classesof p -improperintervalbigraphs. Inchapter4,welookatunitprobeintervalgraphs.Sincetheyhavebeencharacterizedfortreesandbipartitegraphs,wefollowthetechniquesofPrzuljandCorneil forthenextbestcasewhichisfor2-trees.Wecharacterize2-treesthathaveaunit probeintervalrepresentation.Inordertoaccomplishthiswerstcharacterize2caterpillarsandinterior2-caterpillarsintermsofforbiddeninducedsubgraphsand showthat2-treesthatareunitprobeintervalgraphshavetobeinterior2-caterpillars. Atthispointwerealizedthattheproblemwasmuchmoredicultthanweoriginally anticipatedittobe.Sowelookat2-pathsthatareunitprobeintervalgraphsand characterizethem.Thenweusethischaracterizationtondthecharacterizationof interior2-caterpillarwhichareunitprobeintervalgraphs.Finallyweusetheseresults togetacompletecharacterizationof2-treeswhichareunitprobeintervalgraphs. 6

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2.IntervalDigraphs 2.1Introduction Adirectedgraph,ordigraphisagraphinwhicheachedgeisassignedadirection. Anarcdirectededgefromvertex u tovertex v willbedenoted u 7! v; andwewill saythat u beats v .Thesetofverticesofadigraph D willbedenoted V D ,andits sizewillbedenoted j V D j .Asubdigraphof D isadigraphconsistingofasubset of V D ,withonlyarcsfrom D betweentheverticesinthissubset.Throughoutthis paper,wewillonlybeconsideringdigraphsthathaveno2-cyclesorloops;sofortwo vertices u;v 2 V D ,if u 7! v; thenwecannothave v 7! u; or u = v Atournament T isanorientedcompletegraph,soforany x;y 2 V T either x 7! y or y 7! x ,butneverboth,and x 7! x canneverhappen.Anintervaltournamentisatournamentthatisanintervaldigraph.Naturallyasubtournamentof somedigraph D isasubdigraphof D thathappenstobeatournament. Adirectedgraph D isanintervaldigraphifforeachvertex u therecorresponds anorderedpairofintervals S u ;T u suchthatforanytwovertices u;v 2 V D , u 7! v; ifandonlyif S u T v 6 = asshownintheFigure2.1.Intervaldigraphswere introducedin1989bySen,Das,RoyandWest.Theyintroducedintervaldigraphs asananalogueofIntervalGraphs[39]andgaveseveralcharacterizationofinterval digraphsintermsofconsecutiveonespropertyofcertainmatrices,adjacencymatrix andFerrersdigraphs.InthesameyearSen,DasandRoygaveanotheradjacency matrixcharacterizationofspecialtypesofdigraphswhichareveryclosetointerval digraphscalledcirculararcdigraphs[40].Incircular-arcdigraphstherepresentations aremadebyarcsofacircleinsteadofintervalsofrealline.Alsoin1997,Lin,Senand Westgavesomestrongresultsonclassesofintervaldigraphsand0,1-matrices[27].In 1998Westagaingaveamuchshorterversionoftheadjacencymatrixcharacterization ofintervaldigraphs[42].Sofarthereisnoforbiddensubgraphcharacterizationsof intervaldigraphs,butrecently,in2007,intervaltournamentshavebeencharacterized 7

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withacompletelistofforbiddensubtournamentsbyBrown,Busch,andLundgren[6]. Theyshowthatatournamentonnverticesisanintervaldigraphifandonlyifithas atransitive n )]TJ/F15 11.9552 Tf 11.572 0 Td [(1-subtournament.Ifadigraph D isnotatournament,thenitmay beanintervaldigraphevenifitdoesnotcontainatransitive n )]TJ/F15 11.9552 Tf 11.646 0 Td [(1-subtournament asasubdigraph,aslongastherearespecicrestrictionson D . Weexplorewhatrestrictionswecanplaceon D toguaranteethatitisinterval, andinparticularweinvestigateabroaderclassoforientedgraphsonnverticesthat containatransitive n )]TJ/F15 11.9552 Tf 10.593 0 Td [(2-tournamentasasubdigraph.Acompletecharacterization ofintervaldigraphsappearstobereallydicultbutwemanagedtondclassesof orientedgraphsthatareintervaldigraphs.Thusweinvestigatedierenttypesofdigraphsmostofwhichhavecharacteristicsincommonwithtournamentstotryand determineclassesofdigraphsthatcanbeshowntohaveanintervalrepresentation. ForagraphGthatisnotadirectedgraph,anadjacencybetweenvertices u;v 2 V G willbedenotedas u $ v .Abipartitegraph,orbigraph, B isagraphin whichtheverticesarepartitionedintotwosets X and Y ,suchthat X [ Y = V B , andanytwovertices u;v 2 V B canonlybeadjacentifonevertexfrom u , v isin X andtheotherisin Y andthisdoesnotguaranteethattheywillbeadjacent.A bipartitegraphBisanintervalbigraphiftoeachvertextherecorrespondsaninterval suchthattwoverticesareadjacentifandonlyiftheircorrespondingintervalsintersectandeachofthesetwoverticesbelongstoadierentpartitesetasshowninFigure 2.3.In2004,HellandHuanggavesomeinterestingresultsonintervalbigraphs[24]. Moreinterestingworkhasbeendoneonintervalbigraphsandotherrelatedgraphsby BrownandLundgrenin2006[11].Methodsforrecognitionofintervalbigraphsand intervaldigraphshavealsobeeninvestigatedin[31]byMuller.Hegaveadynamic programmingalgorithmrecognizingintervalbigraphsintervaldigraphsinpolynomialtime.Thisalgorithmrecursivelyconstructsabipartiteintervalrepresentation ofagraphfrombipartiteintervalrepresentationsofpropersubgraphs.Wewillsee 8

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laterthatthemodelsforintervaldigraphsandintervalbigraphsarebasicallysame. Weusetheequivalenceofthemodelsforintervaldigraphsandintervalbigraphsin ourinvestigationofwhichorientedgraphsareintervaldigraphs. Anadjacencymatrixofadigraph D ,denoted A D ,isa0,1-matrixthathas a1intheentry a i;j rowi,columnjifandonlyif v i 7! v j forthetwovertices v i ;v j 2 V D .A0,1-matrixhasazeropartitionifafterindependentrowandcolumnpermutationseveryzerocanbelabeledasCorR,suchthatbeloweveryCis anotherCexceptforC'sinthebottomrowandtotherightofeveryRisanotherR exceptforR'sinthefarrightcolumn.Figure2.2givesazeropartitionofamatrix. Wetakeintoaccountimportantresultsonintervaldigraphsandintervalbigraphs andpointoutaconnectionbetweenthetwotoprovethesignicantresultsinthis chapter. Thefollowingthreetheoremsarehelpfultoolsneededtoshowcertaindigraphsare, orarenot,interval. Figure2.1: Intervaldigraph Theorem2.1.1 Sen,Das,Roy,West[39].Adigraph D isanintervaldigraphif andonlyif A D hasazeropartition. Anasteroidaltripleofedges,orATE,isasetofthreeedgesinagraphforwhich thereisapathbetweenanytwooftheseedgesthatavoidstheneighborhoodofthe thirdtheneighborhoodofanedgeisthesetofverticesthatareadjacenttooneof itsend-vertices. 9

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Figure2.2: Zeropartition Figure2.3: Intervalbigraph Theorem2.1.2 Muller[31].IfBisanintervalbigraph,thenthefollowinghold: aBhasnoinducedcycleonmorethan4vertices; bBhasnoasteroidaltripleofedgesinanyinducedsubgraph; cBhasnoinsect2.4asaninducedsubgraph. Adigraph D canberepresentedasabigraph B D bylettingeveryvertex v 2 V D correspondtotwoverticesin B D onefromeachpartiteset x v 2 X and y v 2 Y ,suchthat u 7! v in D ifandonlyif x u $ y v in B ,andthisrelationaccountsfor alltheedgesin B D .Thefollowingtheoremputstheconceptsofintervaldigraphs andintervalbigraphstogethertohelpusidentifydigraphsthathavenointerval representation. Theorem2.1.3 If D isanintervaldigraph,then B D isanintervalbigraph. 10

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Figure2.4: Insect Figure2.5: Bigraphrepresentationofadigraph Wecangetthisresultbyletting I x v = S v and I y v = T v forany v 2 V D where x v and y v arethecorrespondingverticesin B D .So,essentially,themodelsfor intervaldigraphsandintervalbigraphsarethesame. 2.2MotivationfromIntervalTournaments Recently,intervaltournamentshavebeencharacterizedbyBrown,Busch,Lundgren[6]andwewillusepartofamainresultprovedbythemthefollowingtheorem forourproofs. Theorem2.2.1 Brown,Busch,Lundgren[6].SupposethatTisatournamenton nvertices.Thenthefollowingareequivalent: aTisanintervaltournament; 11

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bThasatransitive n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 -subtournament; Thenaturalquestionthatrisesfromthischaracterizationis"Whatarethe othertypesofdigraphsthatcanbeclassiedasinterval?".Sowelookatdigraphs withqualitiesthataresimilartointervaltournaments.Weknowthataninterval tournamentonnverticeshasatransitive n )]TJ/F15 11.9552 Tf 11.616 0 Td [(1-subtournament.Sorstwelookat digraphswiththissamecharacteristicandthisleadsustoageneralizationofTheorem 2.2.1 b a usingasimilarproof.Thefollowingtheoremisageneralizationofthe previoustheorem b a andweusetheconceptofzeropartitionforitsproof. Theorem2.2.2 If D isadigraphonnverticeswithnoloopsor2-cyclesanda transitive n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 -subtournament,then D isanintervaldigraph. Proof: Let D beadigraphonnverticeswithno2-cyclesorloops,andsupposethe inducedsubgraph D )]TJ/F19 11.9552 Tf 12.56 0 Td [(v isatransitivetournamentasshowninFigure2.6.Label theverticesof D )]TJ/F19 11.9552 Tf 12.968 0 Td [(v as v n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ;v n )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 ;:::;v 2 ;v 1 suchthatif v i 7! v j ,then i
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0.AlsonotethatthisdoesnotrequireustochangetheRlabelingofthe0'sfrom A D )]TJ/F19 11.9552 Tf 10.906 0 Td [(v sinceeachofthese0'shada0totherightofit,andwehaveonlypermuted therows. Wecannowlabelthe0'sinthevrowonthebottomasC's,sincenoneofthem willhave1'sbelow.Allofthe0'sin A D havenowbeenlabeledasRorC,sowe haveazero-partition,andTheorem2.1.1impliesthat D isanintervaldigraph. Figure2.6: D )]TJ/F19 11.9552 Tf 11.955 0 Td [(v Notethatifadigraph D onnverticeshasa n )]TJ/F15 11.9552 Tf 12.358 0 Td [(1-subdigraphthatistransitive,itisnotnecessarilyanintervaldigraph.Weonlyknowthat D hasaninterval representationinthiscaseifthetransitive n )]TJ/F15 11.9552 Tf 12.294 0 Td [(1-subdigraphisactuallyatournament.Infact,atransitivedigraphitselfmaynotbeinterval.Anexampleofthisis inFigure2.8,whichshowsatransitivedigraph D thatisnotintervalbyTheorem 2.1.2becauseitsbigraph B D hasaninduced6-cycle. Supposenowforadigraph D ,theinducedsubgraph D )]TJ/F19 11.9552 Tf 12.837 0 Td [(v istheunionofk disjointtransitivetournamentsforsome k 2 N 0 .Wendthattheredoexistseveral suchdigraphsthatarenotintervaldigraphs.Wefound6-cycles,8-cycles,10-cycles 13

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andevenATEsinthecorrespondingbigraphsofsuchdigraphs. InFigure2.9,theinduced10-cycleinthebigraph B D showsthat D isnot anintervaldigraph.However,ithasbeenshowedthatthe10-cycleisthelargest inducedcyclethatcanexistinthebigraph B D ofsuchadigraph D [16].Thus6,8, and10-cyclesaretheonlycyclesinthebigraph B D thatwouldproduceforbidden structures.Itisworthmentioningherethatithasalsobeenprovedthatthelongest inducedpathinthebigraph B T ofanytransitivetournament T consistsofthree edgesandfourvertices[16]. Unfortunately,wealsodiscoveredthattherearemanywaysthatATE'scanappearinthebigraphscorrespondingtodigraphs D inwhich D )]TJ/F19 11.9552 Tf 10.497 0 Td [(v isaunionofdisjoint transitivetournaments.OneexampleisshownbelowinFigure2.7.Weknowthat thatif B D hasanATE,then B D isnotanintervalbigraph,andhence D is notanintervaldigraph.Theseresultsleadustoconsideradierentgeneralization, namely,digraphsDonnverticessuchthattheinducedsubdigraphonn-2vertices givenby D f v 0 ;v 00 g isatransitivetournament. 2.3DigraphswithaTransitive n )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 -subtournament Adigraph D onnverticeswithatransitive n )]TJ/F15 11.9552 Tf 10.234 0 Td [(2-subtournamentcanhavemany structuresinitsbigraph B D thatpreventitfrombeinganintervaldigraph.Figure 2.10belowisanexamplewherewegetaforbiddensubgraphin B D whichisa 6-cycleforwhich D f v 0 ;v 00 g isatransitivetournament.Itisalsoworth-mentioning herethatforthisparticulardigraph D ,addinganycombinationofthedashedarcs ornoneofthemwillresultinthesameinduced6-cyclein B D .Altogether,there areatotalof36possiblecombinationsofthesedashededgesthatwillresultinthis particularinduced6-cycle. TheexampleintheFigure2.10showsthatwecaneasilyendupwithaninduced 6-cyclein B D foradigraph D inwhich D f v 0 ;v 00 g isatransitivetournament. Furthermorethistypeofdigraph D hasmanyforbiddenstructuresthatwillprevent 14

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Figure2.7: ATEinthebigraph itfromhavinganintervalrepresentationincludingmanyinducedcyclesoflengthat least6andmanyATE'sinitsbigraph.Thissuggeststhatperhapsweneedmore restrictionson D tondatypeofdigraphthatweknowisinterval.Thefollowing theoremplacesmorerestrictionsonthearcsdirectedbetweentheverticesofthe transitive n )]TJ/F15 11.9552 Tf 11.955 0 Td [(2-subtournamentandtheothertwovertices v 0 and v 00 . Theorem2.3.1 Let D beadirectedgraphonnverticeswithno2-cyclesorloops, andwithatransitive n )]TJ/F15 11.9552 Tf 12.454 0 Td [(2 -subtournament D = D f v 0 ;v 00 g .Supposeeitheraor bistrue,butnotboth: aSupposenoverticesin D beat v 0 or v 00 ,and v 00 beatsasubsetoftheverticesin D that v 0 beats; bSupposenoverticesin D arebeatenby v 0 or v 00 ,and v 00 isbeatenbyasubsetof theverticesin D thatbeat v 0 . Then D isanintervaldigraph 15

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Figure2.8: 6-cycle Figure2.9: 10-cycle 16

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Figure2.10: Induced6-cycleinthebigraphrepresentation Figure2.11: Morerestrictionsonthearcs Proof: Let D beadigraphonnverticeswithno2-cyclesorloops,andsuppose D = D f v 0 ;v 00 g isatransitivetournament.Thusthesubdigraph D v 00 hasatransitive n )]TJ/F15 11.9552 Tf 10.961 0 Td [(2-subtournament,andbyTheorem2.2.1itisanintervaldigraph.Theorem 2.1.1nowimpliesthat A D )]TJ/F19 11.9552 Tf 12.417 0 Td [(v 00 hasazeropartition.Assumethat D meetsthe conditionsofeitheraorbabove,butnotboth.Thefollowingcasesmustbe considered: i v 0 ,and v 00 havenoarcbetweenthem, ii v 00 hasanarcdirectedto v 0 , iii v 0 hasanarcdirectedto v 00 . 17

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Arrangetherowsandcolumnsof A D )]TJ/F19 11.9552 Tf 12.3 0 Td [(v 00 asintheproofofTheorem2.2.1Now addthecolumnandrowcorrespondingto v 00 atthefarleftwithcolumn v 0 toits rightandbottomofthematrixwithrow v 0 above,respectively.Recallfromthe proofofTheorem2.2.1thatthesubmatrix A D alreadyhaseachofits0slabeled asR. Ifaistruebutbisnot: iLet v 0 ,and v 00 havenoarcdirectedbetweenthem.Row v 0 justaboverow v 00 hasits0slabeledasCs,asintheproofofTheorem2.2.1.Row v 00 willhavea 0ineverycolumnentrythatrow v 0 hasa0,andwillalsohave0sinsomecolumn entriesinwhichrow v 0 has1sand1sintheotherentries.Soeach0inrow v 00 can belabeledasC,whichwilleachbebelowanotherCora1.Also,columns v 0 and v 00 willhaveall0entriessincenoverticesin D beatthesetwovertices,andeachof these0'scanbelabeledasC.Now A D hasazeropartition. iiIf v 00 hasaarcdirectedto v 0 ,then a v 00 v 0 istheonlyentrythatisdierentfrom A D iniabove,anditnowbecomesa1.This1isaproblembecauseitisbelow the0sincolumn v 0 ,whichhadpreviouslybeenlabeledasCs.Sonowmovecolumn v 0 tothefarright.Becausenoverticesfrom D havedirectedarcsto v 0 or v 00 ,the1 atthebottomofcolumn v 0 istheonly1inthatcolumn,andallthe0saboveitcan berelabeledasRs.Now A D hasazeropartition. iiiIf v 0 hasadirectedarcto v 00 ,then a v 0 v 00 istheonlyentrythatisdierentfrom A D iniabove,anditnowbecomesa1.Inthesamemanneraspartii,move column v 00 tothefarrightandrelabelallentriesaboveentry a v 0 v 00 whichareall0s asRs.Thereisalsoone0below a v 0 v 00 incolumn v 00 ,whichcanbelabeledasC.Now A D hasazeropartition. 18

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Ifbistruebutaisnot: iLet v 0 and v 00 havenoarcdirectedbetweenthem.Therowsof A D )]TJ/F19 11.9552 Tf 12.353 0 Td [(v 00 have alreadybeenpermutedasintheproofofTheorem2.2.1sothattherowswitha 1inthecolumn v 0 justrightofcolumn v 00 areatthetop,withallthe0'sinthis columnbelowthe1'sandlabeledasC's.Nowpermutetheserowswith1'sincolumn v 0 sothattherowswith1'sincolumn v 00 areatthetopwith0'sunderneathnote thatcolumn v 0 willstillhaveallits1'saboveits0's.Nowallthe0'sarebelowthe 1'sincolumn v 0 and v 00 ,andeachofthese0'scanbelabeledasC.Therows v 0 and v 00 willbecomposedentirelyof0's,whichcanbelabeledasC's,sincenoverticesin D arebeatenbythesetwovertices.Now A D hasazeropartition. iiIf v 00 hasadirectedarcto v 0 ,then a v 00 v 0 istheonlyentrythatisdierentfrom A D iniabove,anditnowbecomesa1.This1isnowaproblembecauseitis belowthe0'sincolumn v 0 ,whichhadpreviouslybeenlabeledasC's.Sonowmove row v 00 justbelowthelowestrowin A D )]TJ/F19 11.9552 Tf 12.282 0 Td [(v 00 forwhichthereisa1incolumn v 0 . Because v 0 and v 00 donothavedirectedarcstoverticesin D , a v 00 v 0 istheonlyentry inrow v 00 thatisa1,sothe0totheleftincolumn v 00 isstilllabeledasaC,andall the0'stotherightcanbelabeledasR's.Now A D hasazeropartition. iiiIf v 0 hasadirectedarcto v 00 ,then a v 0 v 00 istheonlyentrythatisdierentfrom A D iniabove,anditnowbecomesa1.This1isnowaproblembecauseitis belowthe0'sincolumn v 00 ,whichhadpreviouslybeenlabeledasC's.Nowmoverow v 0 tothetop.Because v 0 and v 00 donothavedirectedarcstoverticesin D , a v 0 v 00 is theonlyentryinrow v 0 thatisa1,soalltheotherentriesare0's,whichareallto therightofentry a v 0 v 00 andcaneachbelabeledR.Nowwehaveazeropartitionfor A D . Ineverycaseabovewehavefoundazeropartitionof A D ,soTheorem2.1.1implies that D isanintervaldigraph. 19

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Itisworthmentioningherethatifthereisnoarcdirectedbetween v 0 and v 00 thenDcanbeprovedtobeanintervaldigraphwithfewerrestrictionsplacedonthe directedarcsbetweentheverticesof D and v 0 , v 00 .Thefollowingtheoremproves thisfact. Theorem2.3.2 Let D beadirectedgraphonnverticeswithno2-cyclesorloops, andwithatransitive n )]TJ/F15 11.9552 Tf 12.473 0 Td [(2 -subtournament D = D f v 0 ;v 00 g ,where v 0 and v 00 are verticesthathavenoarcdirectedbetweentheminD. Suppose v 0 beatsasubsetoftheverticesin D that v 00 beats,or v 00 beatsasubsetofthe verticesin D that v 0 beats.Alsosupposethat v 0 isbeatenbyasubsetofthevertices in D thatbeat v 00 ,or v 00 isbeatenbyasubsetoftheverticesin D thatbeat v 0 . Then D isanintervaldigraph. Proof: Because D isadigraphwithno2-cyclesorloopsthathasatransitive n )]TJ/F15 11.9552 Tf 12.348 0 Td [(2subtournament D = D f v 0 ;v 00 g ,weknowthatthesubgraph D v 00 hasatransitive subtournamentonallbutoneofitsvertices.Theorem2.2.1impliesthat D v 00 is anintervaldigraph,whichisequivalentto A D )]TJ/F19 11.9552 Tf 12.47 0 Td [(v 00 havingazero-partitionfrom Theorem2.1.1. Arrangethesubmatrix A D )]TJ/F19 11.9552 Tf 12.44 0 Td [(v 00 asintheproofofTheorem2.2.1,wherethe rowscorrespondingtoverticesfrom D witharcsdirectedto v 0 areatthetop,sothat allthe0'sincolumn v 0 arebelowthe1'sandeachislabeledasC. Iftheverticesin D withdirectedarcsto v 00 areasubsetofthosewithdirected arcsto v 0 ,thenwecanplacecolumn v 00 totheleftofcolumn v 0 inADandrearrange thetoprowswith1'sincolumn v 0 sothatalltherowswith1'sincolumn v 00 areon top,withthe0'sunderneaththemlabeledasC's.Notethatcolumn v 0 willstillhave allits1'satthetop,with0'sunderneathandlabeledasC's. 20

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Iftheverticesin D witharcsdirectedto v 0 areasubsetofthosewitharcs directedto v 00 ,thenwecanplacecolumn v 00 totheleftofcolumn v 0 again.Allthe rowswith1sincolumn v 0 arealreadyatthetop,whichalsohave1'sincolumn v 00 , andwecanplacetheotherrowswith1'sincolumn v 00 directlybelowtheserowsso thatcolumns v 0 and v 00 havethe0'slabeledasC'sandplacedbelowallofthe1's. Sincewehavearranged A D )]TJ/F19 11.9552 Tf 11.894 0 Td [(v 00 asintheproofofTheorem2.2.1,row v 0 isat thebottomof A D )]TJ/F19 11.9552 Tf 11.955 0 Td [(v 00 andhaseachofits0'slabeledasC. Iftheverticesin D witharcsdirectedfrom v 00 areasubsetofthosewithdirected arcsfrom v 0 ,thenwecanplacerow v 00 belowrow v 0 in A D andlabeleachofits0's asaC,sinceeachoftheseC'swilleitherbebelowa1oranotherC. Iftheverticesin D witharcsdirectedfrom v 0 areasubsetofthosewitharcs directedfrom v 00 ,thenwecanplacerow v 00 justaboverow v 0 in A D andlabeleach ofits0'sasaC,sinceeachoftheseC'swillalwaysbeaboveanotherC. AsintheproofofTheorem2.2.1,therestofthe0'swhicharefromthesubmatrix A D arelabeledasR's.Therefore A D hasazeropartition,andfromTheorem 2.1.1thisimpliesthat D isanintervaldigraph. Theorem2.3.2willfailtoholdifthereexistanarcbetween v 0 and v 00 .Thegure 2.14showsaninduced6-cyclein B D correspondingtothedigraph D whichhasarc v 0 7! v 00 .In D , v 00 beatsasubsetoftheverticesin D -[ v 0 , v 00 ]that v 0 beatsand v 00 is beatenbyasubsetoftheverticesinD-[ v 0 , v 00 ]thatbeat v 0 . 21

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2.4Morethanonenontrivialstrongcomponent Astrongcomponentofadigraph D isamaximalstronglyconnectedsubgraph. Adirectedgraphisstronglyconnectedifforeverypairofvertices v , u thereexists atleastonepathfrom v to u ,andatleastonepathfrom u to v .Brown,Busch, Lundgren[6]provedthefollowingtheoremaboutstrongcomponentsintournaments. Theorem2.4.1 Notournamentthathastwoormorenon-trivialstrongcomponents isanintervaltournament. However,anintervaldigraphcanhavemorethanonenontrivialstrongcomponent.Infact,certaindigraphswithtwoormorenontrivialstrongcomponentsthat havespecictypesofadjacencymatricesalwayshaveanintervalrepresentation,as thefollowingtheoremshows.Figure2.15showsanintervaldigraphwith2non-trivial strongcomponents.Wewillrequireacoupleofnewdenitionstoprovethatcertain digraphswithspecicadjacencymatricesandtwoormorenon-trivialstrongcomponentswillalwaysbeinterval. Denition2.4.2 AnRzeropartitionofa f 0,1 g -matrixisazeropartitionin whicheachofthe0'softhematrixcanbelabeledasRwheretotherightofeach RisanotherRaftercolumnpermutations.Likewise,aCzeropartitionisazero partitioninwhicheachofthe0'scanbelabeledasCwherebeloweachCisanother Cafterrowpermutations. Thefollowingtheoremshowsthatadigraphwithexactlytwonon-trivialstrong componentswillbeintervaliftheadjacencymatricesofthestrongcomponentsand theadjacencymatrixdeterminingtherelationbetweenthestrongcomponentsare specic.Itisworthmentioningherethatthefollowingresulthasbeengeneralizedto adigraphwith k 2 N non-trivialstrongcomponentsin[16]. 22

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Theorem2.4.3 Let D beadirectedgraphonnverticeswithexactlytwonontrivial strongcomponents, D 1 and D 2 .Supposethatasubsetoftheverticesof D 1 havearcs directedtoasubsetoftheverticesof D 2 butnoverticesin D 2 havearcsdirectedto verticesin D 1 . Ifthesubmatrices A 1 = A D 1 and A 2 = A D 2 bothhaveaCzeropartition, andthesubmatrix B whoserowsrepresenttheverticesof D 1 ,andwhosecolumns representtheverticesof D 2 hasanRzeropartition,thenDisanintervaldigraph. Proof: Lettheadjacencymatrix A D bearrangedasfollows: A D = 0 B @ A 1 B 0 A 2 1 C A Ourhypothesisstatesthatnotonlydoes D containexactlytwonontrivialstrong components D 1 and D 2 ,buttheiradjacencymatrices, A 1 and A 2 ,musthaveaC zeropartitionafterrowpermutations.Also,thearcsdirectedfromverticesin D 1 to verticesin D 2 mustbesuchthattheadjacencysubmatrix B whoserowsrepresent theverticesof D 1 ,andwhosecolumnsrepresenttheverticesof D 2 musthaveanR zeropartitionaftercolumnpermutations.Assumingthattheseconditionsaremet, thecolumnpermutationsusedtogetanRzeropartitionof B whichdonotaect theorderoftheentriesinthecolumnsof A 1 or A 2 willnotchangetheClabelingsof the0sin A 1 and A 2 ,andtherowpermutationsusedtogetaCzeropartitionof A 1 and A 2 whichdonotaecttheorderoftheentriesintherowsof B willnotchange theRlabelingsofthe0sin B .Therefore A D hasazeropartitioninwhicheach ofthe0sinthesubmatricesrepresentedby A 1 , A 2 ,and0arelabeledasC,andthe 23

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submatrix B haseachofits0slabeledasR.ThisimpliesbyTheorem2.1.1that D isanintervaldigraph. Wehavestudiedthestructuresofseveraldigraphsinordertoascertainwhen theyareinterval.Wefocussedondigraphsthatpossesssomesimilarqualitiesof intervaltournamentssuchasthedigraphswithatransitivesubtournamentonallbut twoverticesandthedigraphsforwhichtheremovalofonevertexleavesasubdigraph thatisatransitivesub-tournament.Thesetypesofdigraphsseemedtobethenatural placetostartbecausethereisacompletecharacterizationofintervaltournaments. Wesuccessfullyfounddierentrestrictionsthatforceadigraphtobeintervalthus ndingsomespecialclassesofintervaldigraphs. Therearestillnumerousclassesofdigraphsthatwillbeintervalwithsomerestrictionsontheirarcswhichhavenotbeenexploredyet.Surelytheproblemof ndingacompleteforbiddensubdigraphcharacterizationofintervaldigraphswillbe averydicultonesincethelistofforbiddenstructuresisalreadyverylong.Much workstillneedstobedoneonthistopic. 24

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Figure2.12: Exampleofadigraphsatisfyingaconditionofthetheorem2.3.1and itszeropartition 25

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Figure2.13: Exampleofadigraphsatisfyingaconditionofthetheorem2.3.2and itszeropartition Figure2.14: Exampleofadigraphthatfailsthehypothesisoftheorem2.3.2since v 0 beats v 00 andhenceformsa6-cycleinitsbigraphrepresentation Figure2.15: Strongcomponents 26

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3.IntervalBigraphImpropriety 3.1Introduction Anintervalgraphisproperifandonlyifithasarepresentationinwhichno intervalcontainsanother.BeyerlandJamisonintroducedthestudyof p -improper intervalgraphswherenointervalcontainsmorethan p otherintervalsin2008.Thus aproperintervalgraphisa0-improperintervalgraph.Inthischapterweextendthe ideabyintroducing p -improperintervalbigraphs,wherenointervalcontainsmore than p otherintervalsofthesamepartiteset.Severalauthorshavestudiedproper intervalbigraphs.Oneofthesecharacterizationshasthreeforbiddensubgraphs.We ndboundsonthestructureof p -improperintervalbigraphsandcharacterizeaspecial caseof p -improperintervalbigraphs. Letagraph G havevertexset V andedgeset E . x;y 2 V beingadjacentis denotedby xy 2 E .Anitesimplegraph G V;E isanintervalgraphifwecannd amapping : v )167(! I v fromverticesofGtointervalsontherealline,suchthatthe edge xy existsifandonlyif I x I y 6 = forall x;y 2 V G .Intervalgraphswererst discussedbyHajos[22].Thestudyofintervalgraphsalsohasitsorigininapaperof Benzer[1]whowasstudyingthestructureofbacterialgenes. Awell-knowncharacterizationofintervalgraphswasgivenbyLekkerkerkerand Bolandin1962.TheLekkerkerker-BolandTheorem[26]saysthatchordlesscycles andasteroidaltriplesformadeningclassofforbiddensubgraphsfortheclassof intervalgraphs.An asteroidaltriple ATin G isaset A ofthreeverticessuchthat betweenanytwoverticesin A thereisapathbetweenthemthatavoidsallneighbors ofthethird.AnaturalextensionofATisanAsteroidalTripleofEdges.An asteroidal tripleofedges ATEisasetofthreeedgessuchthatforanytwothereisapath fromthevertexsetofonetothevertexsetoftheotherthatavoidstheneighborhood ofthethirdedge.Anaturalextensionofintervalsgraphs,calledintervalbigraphs, wereintroducedbyHarary,Kabell,andMcMorris[23]in1982.Abipartitegraph 27

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G X;Y;E isanintervalbigraphiftoeveryvertex, v 2 V G ,wecanassignan intervaloftherealline, I v ,suchthat xy 2 E G ifandonlyif I x I y 6 = ; and x 2 X and y 2 Y .Thesegraphshavebeenstudiedbyseveralauthors[13],[27].Todateno forbiddensubgraphcharacterizationofintervalbigraphshasbeenfound,butinitially itwasthoughtthatasteriodaltriplesofedgesalongwithinducedcycleslargerthan 4wouldwork[23].Theyprovedthatif B isanintervalbigraphthen B doesnot haveATE.However,Muller[31]foundinsectsandHellandHuang[24]foundedge asteriodsandbugsasforbiddensubgraphs,andtodateacompletecharacterization isstillnotavailable. FredRoberts[35]in1969characterizedproperintervalgraphs.Properinterval graphsaregraphswhichhaveanintervalrepresentationsuchthatnointervalcontainsanother.Anintervalgraphisproperifandonlyifitdoesnotcontain K 1 ; 3 as aninducedsubgraph.Unitintervalgraphsarethegraphshavinganintervalrepresentationinwhichalltheintervalshavethesamelength.Claw-freeintervalgraphs aretheintervalgraphswithoutaninducedcopyoftheclaw, K 1 ; 3 .Robertshasalso shownthattheclassofproperintervalgraphscoincidewiththeclassesofunitinterval graphsandclaw-freeintervalgraphs. Properintervalbigraphsarebigraphswhichhaveanintervalrepresentations wherenointervalcontainsanother.Severalcharactizationsofproperintervalbigraphshavebeenfoundinthelastdecade.ThegraphsinFigure3.4aretheforbidden subgraphsforproperintervalbigraphsfoundbyLinandWest[27]. Theideaofproperintervalgraphswasnaturallyextendedto p -properinterval graphsbyProskurowskiandTelle[33].The p -properintervalgraphsaregraphswhich haveanintervalrepresentationwherenointervalisproperlycontainedinmorethan p otherintervals.BeyerlandJamison[3]investigatedavariationincontainmentand introduced p -improperintervalgraphswherenointervalcontainsmorethan p other intervals.A p -improperrepresentationofagraphisanintervalrepresentationof 28

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thegraphwherenointervalcontainsmorethan p otherintervals.Figure3.2isan illustrationofa1-improperintervalgraph.Thusany0-improperintervalgraphisa properintervalgraph.ItcanbeeasilycheckedinFigure3.1that K 1 ; 3 isa1-improper intervalgraph. Inthischapterwegeneralizetheclassof p -improperintervalgraphsto p -improper intervalbigraphs.WelookatideaswhichBeyerlandJamisoninvestigatedforinterval graphsandapplytheminintervalbigraphs.Somedenitionshadtobemodiedsince weareworkingwithtwopartitesets. A p -improperintervalbigraphisanintervalbigraphwherenointervalcontains morethan p intervalsfromthesamepartiteset.Inordertobeabletogeneralizethe ideasfor p -improperintervalgraphsto p -improperintervalbigraphs,werestrictthe containmentofintervalstoanindividualpartitesetasshownintheFigure3.3.Beyerl andJamisoninvestigatedideassuchasimpropriety,weight, p -criticalandbalancefor intervalgraphs,andweinvestigatethesameideaswithsomemodieddenitionsfor intervalbigraphs.Inthischapterwendrestrictionsinthestuctureofaninterval bigraphforittobea p -improperintervalbigraph.Wealsostudyspecialclassesof p -improperintervalbigraphs. Figure3.1: K 1 ; 3 3.2ImproprietyandWeightofIntervalBigraphs Fromthispointon B = B X;Y;E willdenoteanite,connected,intervalbigraphwithbipartition f X;Y g andthesets X and Y willbereferredtoasthepartite 29

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Figure3.2: p -ImproperIntervalGraph Figure3.3: 0-ImproperIntervalBigraph sets.A0-improperintervalbigraphisaproperintervalbigraph.Thefollowing propositionjustiesourwayofdeninga p -improperintervalbigraph.Itexplains thesuciencyofrestrictingtheinclusionofintervalstoasinglepartitesetinan intervalbigraph. Proposition3.2.1 B isaproperintervalbigraphifandonlyifithasaninterval representationwherenointervalfromapartitesetcontainsanotherintervalfromthe samepartiteset. Proof: Suppose B X;Y;E isaproperintervalbigraph.Itfollowsfromthe denitionthatithasanintervalrepresentationwherenointervalfromapartite setcontainsanotherintervalfromthesamepartiteset,sincenointervalcontains another. Suppose B hasanintervalrepresentationwherenointervalfromapartiteset 30

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containsanotherintervalfromthesamepartiteset.If B isnotproperthenithas oneofthegraphsinFigure3.4asaninducedsubgraph[27].Itiseasytocheckfrom theintervalrepresentationofB1,B2andB3illustratedinFigure3.4andFigure 3.5thattheyallareforcedtohaveanintervalwhichcontainsanotherintervalfrom thesamepartiteset.InFigure3.4weavoidthecontainmentofanintervalfrom Y inanotherintervalfrom Y .Thisforcesthecontainmentofanintervalfrom X in anotherintervalfrom X .InFigure3.5weavoidthecontainmentofanintervalfrom X inanotherintervalfrom X .Thisforcesthecontainmentofanintervalfrom Y or X inanotherintervalfrom Y or X .B1istheonlyexamplewhereweneverattain acontainmentfrom Y .Italwayshasanintervalfrom X containinganotherfrom X .Thuswecannotavoidan X intervalproperlycontaininganotherfrom X ora Y intervalproperlycontaininganotherfrom Y .Hence B hasanintervalwhichcontains anotherintervalfromthesamepartitesetwhichcontradictsourassumption. Figure3.4: ForbiddensubgraphsforproperIntervalbigraphwiththeirinterval representationstoforcecontainmentfrom X Figure3.5: IntervalrepresentationsofB1,B2,B3toforcecontainmentfrom Y 31

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FirstwewillgeneralizethenotionofimproprietyintroducedbyBeyerlandJamison.Theideaofimproprietycomesfrom p -improperintervalbigraphs. Denition3.2.2 The improprietyofavertex z 2 X withrespecttotheinterval representation isthenumberofvertices x 2 X suchthat I x I z ,whichwewill denoteas imp z . Denition3.2.3 The improprietyofarepresentation isthemaximum imp z forall z 2 X;Y whichwewilldenoteas imp B . Denition3.2.4 The improprietyoftheintervalbigraph B istheminimum imp B overallpossibleintervalrepresentationswhichwewilldenoteas imp B .Arepresentationwhichgivestheminimumimproprietywillbecalledaminimalrepresentation. Anyintervalbigraphcanhaveaninnitenumberofintervalrepresentations.So itiseasytoseethattheimproprietyistheleast p forwhichthegraphisa p -improper intervalbigraphasshownintheFigure3.6.Figures3.8and3.9giveexamplesof graphswhoseimproprietyis1and2respectively.Itiseasytocheckthat I z I x ;I y isforced. Denition3.2.5 For z 2 B X;Y;E ,acomponentof B )]TJ/F19 11.9552 Tf 12.418 0 Td [(z willbecalleda local component of z .Alocalcomponentof z is exterior ifandonlyifitcontainsanedge xy suchthat x;y= 2 N z .SeeFigure3.7. Itisworthmentioningherethatallthebigraphsherehaveconnectivityone. Thefollowinglemmagivesusaboundonthenumberofexteriorlocalcomponents anyvertexofanintervalbigraphcanhave. Lemma3.2.6 Avertex z inanintervalbigraphcanhaveatmosttwoexteriorlocal components. 32

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Figure3.6: Improprietyof imp B 1 =2, imp B 2 =1 Figure3.7: Exteriorlocalcomponents Proof: Wewillprovethisbycontradiction.Letusassumethat B X;Y isa connectedintervalbigraphand z 2 X isavertexin B withatleast3exteriorlocal components.Sothereexists3edges x 1 y 1 ;x 2 y 2 ;x 3 y 3 in3oftheseexteriorcomponents suchthat x 1 ;y 1 ;x 2 ;y 2 ;x 3 ;y 3 = 2 N z .Wecanndapathbetweenanytwopairs oftheseedges,sayfrom x 1 y 1 to x 2 y 2 through z whichavoidstheneighborhoodof x 3 y 3 since x 3 ;y 3 = 2 N z .Hencetheedges x 1 y 1 ;x 2 y 2 ;x 3 y 3 formanasteroidaltriple ofedgeswhichisaforbiddensubgraphforanintervalbigraph[23].Thusweget acontradiction.Henceanyvertex z inanintervalbigraphcanhaveatmosttwo exteriorlocalcomponents. 33

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Note: Let B X;Y beanintervalbigraph.If C i isanexteriorcomponentofany vertex z 2 X thenintheintervalrepresentationthereexistavertex y 2 C i Y such that I y I z = .So C i mustalwaysbeonthesideof I z . Figure3.8: Intervalbigraphwithimpropriety1 Figure3.9: Intervalbigraphwithimpropriety2 Thefollowingdenitionandtheoremleadstoaboundontheimpropriety. Denition3.2.7 Let B X;Y;E beanintervalbigraph.Let z 2 X and C 1 ;C 2 ;::::;C n bethelocalcomponentsof z orderedfromlefttorightthelefthandcomponents haveintervalsonthelefthandsideoftheintervalrepresentationandtherighthand componentswillhaveintervalsontherighthandsideoftheintervalrepresentation suchthat C 1 and C n arechoseneithertobeexteriorcomponentsiftheyexistor non-exteriorlocalcomponentssuchthat j C 1 X jj C i X j ;i =2 ;::: n )]TJ/F15 11.9552 Tf 12.707 0 Td [(1 and j C n X jj C i X j ;i =2 ;::: n )]TJ/F15 11.9552 Tf 11.959 0 Td [(1 .The weightofthevertex z isthetotalnumber 34

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ofverticesin C 2 X [ :::: [ C n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 X ,whichwewilldenoteas wt z .Wewill call C 2 ;C 3 ;::::;C n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 thesmallernon-exteriorlocalcomponentsof z .The weight of X isthemaximumoftheweightsofallthevertices x 2 X ,whichwewilldenoteas wt X .Theweightofavertexin Y andtheweightof Y aredenedsimilarly.The weightofanintervalbigraph , wt B ,isthemaximumoftheweightsofitstwopartite sets. ThegraphsinFigure3.9andFigure3.11haveweights2eventhoughtheyhave adierentnumberofexteriorcomponents.Thisisbecauseweconsiderthesmallest non-exteriorlocalcomponentstocalculatetheweights. Figure3.10: Calculationofweight Figure3.11: Intervalbigraphwithweight2 Thenexttheoremgivesusaboundontheimpropriety. Theorem3.2.8 Let B X;Y;E beanintervalbigraph.Let z 2 B .Thentheimproprietyof B isatleasttheweightof z . Proof: Let beaminimalintervalrepresentationoftheintervalbigraph B . Let z 2 X and C 1 ;C 2 ;::::::C n bethelocalcomponentsof z orderedfromlefttoright 35

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asinDenition3.2.7.Let I z denotetheintervalof z .So wt z = P i =2 ;::: n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 j C i X j . Theintervalsofverticesfrom X in C 2 ;::::::;C n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 mustbeinside I z becausethereare vertices y 1 2 C 1 ;y n 2 C n thatarenotadjacenttoanyvertexfrom X in C 2 ;::::::;C n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 butareadjacentto z .Hence,forany andforany z 2 Xwt z imp z .Since imp B =max[ imp z i forall z i 2 B ], imp z imp B ,andso wt z imp B . Hencetheimproprietyof B isatleasttheweightof z . Corollary3.2.9 Foranyintervalbigraph B X;Y;E , imp B wt B . Proof: Theprooffollowsdirectlyfromtheprevioustheoremanddenitions. Note :Nowthatwehavetheaboveboundonweight,thereasonbehindthe restrictionofthecontainmenttoasinglepartitesetcanbediscussedmoreexplicitly. Ifwedidnotconneittoanindividualpartiteset,theaboveresultwouldfailto holdforintervalbigraphs.Forexample,consider K 1 ; 4 .Thisisanintervalbigraph. Letuscallit B X;Y;E .Withoutlossofgenerality,thereisonlyonevertexin X andtheremainingfourverticesbelongto Y .Byourdenitionofimproprietyand weight,wehavethe wt X =0= imp B .Thustheabovebound wt X imp B holdshere.Ifwehadnotrestrictedtheimproprietytoasinglepartitesetthenthe denitionofweightwouldalsobedierent.Weightalsowouldnothavebeenconned toanindividualpartiteset.Itwouldtakeintoaccountalltheverticesinthe n )]TJ/F15 11.9552 Tf 12.181 0 Td [(2 smallestnonexteriorcomponentsofanyvertexwith n localcomponents.Therefore imp B =0and wt X =2whichisillustratedinFigure3.12.Thisdeesthebound wt X imp B .Thisboundholdsforintervalgraphsandsinceitiseasierto calculate wt B than imp B ,wewantedthisusefulboundhere. 3.3 p -criticalintervalbigraphs Wehaveseenincorollary2.9thattheweightofanypartitesetofaninterval bigraphcanbeatmosttheimproprietyofthebigraph.Figure3.13andgure3.14 36

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Figure3.12: Changesinweightduetovariationindenition Figure3.13: Intervalbigraphwhere wt X =0
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Figure3.14: Intervalbigraphwhere wt X =1= imp B =1 Notethatinanybalancedpartitesetofanintervalbigraphwith imp B > 0the basepointmusthaveatleast3localcomponents.Otherwisetheweightbecomes0 andhencetheimproprietyis0,leadingtoacontradiction.Moreoverifthe wt X > 0 thenthebasepointof X musthaveatleast3localcomponentswithverticesfrom X inallthreecomponents. Denition3.3.2 Apartiteset X ofanintervalbigraph B X;Y;E is p -critical with respecttoimproprietyifandonlyif B hasimpropriety p ,andtheremovalofany vertex z 2 X decreasestheimproprietyof B . Thebalanceandcriticalityofdierentpartitesetsmightnotbethesameforan intervalbigraph.InFigure3.15wehaveillustrationsofbalanceandcriticalityofthe partitesetsofvariousintervalbigraphs.Weprovelaterinthissectionthatboththe partitesetsofanintervalbigraphcannotbebalancedand p -criticalatthesametime. Wewillnowfocusourattentiononaspecialclassofgraphs.Fortheremaining partofthissectionwewillconsiderintervalbigraphswithabalancedand p -critical partiteset.Withthisrestrictiononapartitesetofanintervalbigraphwewillprove interestingresultsaboutthestructuresofthisparticularclassofbigraphs.Thenext theoremsandlemmasgiveusanideaaboutthestructureofanintervalbigraphwith abalancedand p -criticalpartiteset. 38

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Figure3.15: Illustrationsofbalanceand p -criticality Theorem3.3.3 Let z beabasepointofabalanced p -criticalpartiteset X ofthe Bigraph B X;Y .IfCisanexteriorlocalcomponentof z ,thenChasexactlyone vertexfromX. Proof: Let B X;Y beanintervalbigraph.Suppose X isbalancedand p criticaland z isitsbasepoint.Wewillprovethisbycontradiction.Weusetechniques similartothoseusedbyBeyerlandJamison[3].LetCbeanexteriorlocalcomponent of z thathasatleast2vertices x;v 2 X .Let x betheclosestvertexto z suchthat itisonanedge xy and y= 2 N z .Figure3.16illustratespossiblepositionsofthe vertex x .Let H bethegraphobtainedfrom B X;Y bydeletingallthevertices fromCexceptthoseonthepathfrom z to y thatcontains x .Bytheorem3.2.8 wt H z imp H .Since X is p -critical, imp H
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non-exteriorcomponentsremainthesame,whichimplies wt X z = wt H z : Hence wt H z imp H 0thenabasepointhasatleast 3localcomponentswithverticesfrom X inallthreecomponents.Dependingon dierentscenarios,thelocalcomponentscanbeatthesideoftheinterval I z ormust betotallycontainedinit.Inanyintervalrepresentation,wewillcalltheleftmost andrightmostendlocalcomponentsofanyvertexitslocalsidecomponents.The remainingnon-sidelocalcomponentswillbecalledthelocalinnercomponents.In Denition3.2.7, C 1 and C n arealwaysthelocalsidecomponents.Iftheyareexterior localcomponentsthentheywillcontainverticesfrom Y non-adjacentto z andso theymustbeonthesidesof I z .Iftheyarenotexteriortheninordertoobtaina minimalrepresentationtheymustagainbeonthetwosidesof I z .Wewillcall C 1 theleftmostlocalcomponentand C n therightmostlocalcomponent. 40

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Lemma3.3.4 Anybalancedand p -critical p> 0 partitesetofanintervalbigraph B X;Y;E hasexactlyonebasepoint. Proof: Let X beabalancedandp-criticalpartitesetofanintervalbigraph B X;Y;E .Weneedtoshowthat X hasonlyonebasepoint.Wewillprovethisby contradiction.Letusassumethat X hastwobasepointsnamely x and v .Since X is balancedand p -critical p> 0,both x and v musthaveatleast3localcomponents withverticesfrom X .Since x isabasepoint,ithasatleast3localcomponentsand v mustbeinoneofthem.Thisleadstothefollowingcases: Case1:If v 2 CwhereCisanexteriorlocalcomponentof x ,then,byTheorem 3.3.3, v istheonlyvertexfrom X inC.Thisimpliesthat v canhaveexactlyonelocal componentcontainingverticesfrom X .Thisisacontradiction. Case2:If v 2 CwhereCisanon-exteriorlocalinnercomponentof x thenevery vertexfrom Y whichisadjacentto v isalsoadjacentto x sinceCisnotexterior. Thus B v isconnected.So v hasjustonelocalcomponentwhichisacontradiction. Case3:If v 2 CwhereCisanon-exteriorlocalsidecomponentof x ,everyvertex from Y whichisadjacentto v isalsoadjacentto x becauseCisnotexterior.Wecan concludethat v againhasjustonelocalcomponent,whichisacontradiction. Thefollowingtheoremprovesthatboththepartitesetsofanintervalbigraph cannotbebalancedand p -criticalatthesametime. Theorem3.3.5 LetBX,Ybeanintervalbigraph.BothXandYcannotbebalanced and p -critical p> 0 atthesametime. Proof: Wewillprovethisbycontradiction.Letusassumethatboth X and Y are balancedand p -critical.Let v and w bethebasepointsof X and Y respectively. Since X isbalancedand p -criticaland p> 0,itsbasepointmusthaveatleast3local 41

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componentswithverticesfrom X .Let C 1 ;C 2 ;::C j ;::::C n be n localcomponentsof v orderedfromlefttorightand n 3.Hence v isadjacenttoatleast n verticesfrom Y .Since B isconnected, w mustbeinsomelocalcomponentof v .Let w 2 C k ;k 2 1 ; 2 ;:::n .Since Y isbalancedand p -critical, w musthaveatleast3localcomponents withonelocalcomponentcontaining v [ C i , i =1 ;:::n and i 6 = k asaninduced subgraph,whichhasatleast n )]TJ/F15 11.9552 Tf 10.742 0 Td [(1verticesfrom Y .Letuscallthislocalcomponent ofw A 1 where j A 1 Y j n )]TJ/F15 11.9552 Tf 10.949 0 Td [(1.Since n 3thereareatleast2localcomponents of v say C j ;C m and j;m 6 = k ,eachofwhichcontainsavertexfrom X notadjacentto w .Since C j and C m arecontainedin A 1 ,itimpliesthat A 1 isanexteriorcomponent of w .Since A 1 isanexteriorcomponentof w itcancontainexactlyonevertexfrom Y [Theorem3.3.3].But j A 1 Y j n )]TJ/F15 11.9552 Tf 12.513 0 Td [(1and n 3whichisacontradiction. Henceboth X and Y cannotbebalancedand p -criticalatthesametime. Nextwedeterminethegeneralstructureofanintervalbigraphthathasabalancedand p -criticalpartiteset.Thefollowinglemmaandtheoremgiveanexplicit ideaaboutthecardinalityofthelocalsidecomponentsofthebasepointofanybalancedand p -criticalpartitesetofanintervalbigraph.Thelemmagivesusthecardinalityandthefollowingtheoremusesthelemmatodemonstratethestructure.If arepresentationhasbeengiven, l v and r v willdenotetheleftandrightendpoints, respectively,oftheinterval I v representinganyvertex v . Lemma3.3.6 Let X beabalancedandp-criticalpartitesetofanintervalbigraph B X;Y and v 2 X bethebasepointof X withatmostoneexteriorcomponent.Let C 1 ;C 2 ;::::::::C n belocalcomponentsof v orderedfromlefttorightasinDenition 3.2.7.Let beaminimalintervalrepresentationsuchthat C 1 isanon-exteriorlocal sidecomponentof v in and C n istheotherlocalsidecomponent.If C j isalocal innercomponentof v suchthat j C j X j islargestamongalltheotherlocalinner componentsof v ,then j C j X j = j C 1 X j if C n isexteriorand j C j X j = j C 1 X j 42

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= j C n X j if C n isnon-exterior. Proof: Wewillprovethisbycontradiction.Let v 2 X bethebasepointof X withthe correspondingminimalintervalrepresentation .Thelocalcomponentsof v with respecttotheminimalintervalrepresentation are C 1 ;C 2 ;::::::::C n orderedfromleft torightwith C 1 asthenon-exteriorlocalsidecomponentand C n astheotherlocal sidecomponent.Firstlet C n beanexteriorlocalcomponentandpick j suchthat j C j X jj C i X j forall i =2 ; 3 ;:::: n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1. FromDenition3.2.7, j C 1 X jj C j X j .Assumethat j C 1 X j > j C j X j , then j C 1 X j > j C i X j ;i =2 ; 3 ;:::: n )]TJ/F15 11.9552 Tf 12.198 0 Td [(1.Let x 1 2 C 1 X andlookat B )]TJ/F19 11.9552 Tf 12.198 0 Td [(x 1 . Since C 1 isnon-exterior,everyvertexfromYin C 1 adjacentto x 1 isadjacentto v . So B )]TJ/F19 11.9552 Tf 12.166 0 Td [(x 1 isstillconnected.Since j C 1 X j > j C i X j ;i =2 ; 3 ;:::: n )]TJ/F15 11.9552 Tf 12.166 0 Td [(1,removal of x 1 from C 1 doesnotaecttheweightof X .Hence wt X )]TJ/F19 11.9552 Tf 11.802 0 Td [(x 1 = wt X .Since X is p -critical,removalofanyvertexfrom X willreducetheimproprietyof B byone. Thus imp B )]TJ/F19 11.9552 Tf 11.418 0 Td [(x 1 0.Itprovesthat B canhaveexactlythree structuresdependingonthenumberofexteriorcomponents. Theorem3.3.7 Let B X;Y;E beanintervalbigraph.If X isbalancedand p -critical 43

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Figure3.17: Possiblestructuresofanintervalbigraphwithabalancedand p -critical partiteset then B hasoneofthefollowingstructures: i B ' B 1 where B 1 X;Y isanintervalbigraphand v isthebasepointof X which hasnlocalnon-exteriorcomponents C 1 ;C 2 ;::C i ;::::C n orderedfromlefttorightas inDenition3.2.7suchthatthereareatleast3localcomponents C 1 ;C j ;C n where j C j X j = j C 1 X j = j C n X jj C i X j ;i =2 ;:::::; n )]TJ/F15 11.9552 Tf 12.066 0 Td [(1 .Furthermoreeachof C 1 and C n hasavertexfrom Y whichisadjacenttoallverticesfrom X in C 1 and C n . 44

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ii B ' B 2 where B 2 X;Y isanintervalbigraphand v isthebasepointof X which hasexactlyoneexteriorcomponent C n and n )]TJ/F15 11.9552 Tf 12.855 0 Td [(1 localnon-exteriorcomponents C 1 ;C 2 ;::C i ;::::C n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 orderedfromlefttorightasinDenition3.2.7suchthatithasat least2localcomponents C 1 ;C j where j C j X j = j C 1 X jj C i X j ;i =2 ;::::: n )]TJ/F15 11.9552 Tf 10.462 0 Td [(1 . Furthermoretheleftmostlocalcomponent C 1 hasavertexfrom Y whichisadjacent toallverticesfrom X in C 1 . iii B ' B 3 where B 3 X;Y isanintervalbigraphand v isthebasepointof X which has C 1 ;C 2 ;::C i ;::::C n localcomponentsorderedfromlefttorightwithexactlytwo exteriorcomponents C 1 ;C n andtheremaining n )]TJ/F15 11.9552 Tf 12.857 0 Td [(2 componentsarelocalinner components. Proof: B X;Y;E isanintervalbigraphwhere X isbalancedand p -critical. Weneedtoprovethat B isisomorphicto B 1 ;B 2 or B 3 dependingonthenumber ofexteriorcomponentsthebasepointof X has.Let v 2 X bethebasepointof X with n localcomponents C 1 ;C 2 ;::C i ;::::C n orderedfromlefttoright.ByLemma 3.2.6weknowthat v canhaveatmost2exteriorcomponentsandthisgivesus3cases. CaseI:If v 2 X hasnoexteriorcomponentsthenallthecomponentsof v arenonexteriorlocalcomponents.HencebyLemma3.3.6theremustbeatleast3components C 1 ;C j ;C n where j C j X j = j C 1 X j = j C n X jj C i X j ;i =2 ;::::: n )]TJ/F15 11.9552 Tf 11.904 0 Td [(1.Thus B ' B 1 . Letusnowanalyzethelocalsidecomponent C 1 .Since C 1 isnotanexteriorcomponent of v 2 X ,itcanhavemorethanonevertexfrom X .Sincetheweightof v isdetermined bythesmallest n )]TJ/F15 11.9552 Tf 12.959 0 Td [(2localinnercomponentsof v andbythepreviousLemma j C 1 X j = j C j X jj C i X j forall i =2 ; 3 ;:::: n )]TJ/F15 11.9552 Tf 11.443 0 Td [(1,thecomponent C 1 doesnot contributeanythingtowardstheweightof v .Letusassumethat j C j X j = k .Hence j C 1 X j = k .Since X isbalanced, p -criticalwith p> 0, C 1 musthaveverticesfrom 45

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both X and Y .Let x 1 ;x 2 ;::::::::x k 2 C 1 .Therecanbemorethanonevertexfrom Y in C 1 .Since C 1 isanon-exteriorcomponent, I y j I v 6 = forall y j 2 C 1 *. Since X isbalanced,no x i contributestotheimproprietyof v for x i 2 C 1 ,which impliesthatnoneofthe x i ;i =1 ; 2 ;::::n arecontainedin I v **. Since C 1 istheleftmostlocalcomponent,itimpliesthat l x i l v .from**forall j Takethe x i suchthat r x i istheleast.This x i mustbeadjacenttosome y j which isadjacentto v .Thus I y j I x i 6 = ,andbyourchoiceof x i , I y j I x m 6 = forall x m 2 C 1 .So y j isadjacenttoallverticesfrom X in C 1 .Thisstructureisillustrated inFigure3.17.Similarly C n alsohasavertexfrom Y whichisadjacenttoallvertices from X in C n . CaseII:If v 2 X hasexactly1exteriorcomponentthenithasanon-exterior sidecomponent.Let C n betheexteriorcomponent.AgainbyLemma3.3.6itmust haveatleast2localcomponents C j and C 1 suchthat j C 1 X j = j C j X jj C i X j forall i =2 ::::; n )]TJ/F15 11.9552 Tf 12.199 0 Td [(1andoneofthese,say C 1 ,mustbethelocalsidecomponent. Thus B ' B 2 .ByargumentssimilartothoseusedinCaseI, C 1 alsohasavertex from Y whichisadjacenttoallverticesfrom X in C 1 .ThisisillustratedinFigure3.17. CaseIII:If v 2 X has2exteriorcomponents C 1 and C n thenalltheother n )]TJ/F15 11.9552 Tf 10.857 0 Td [(2 localcomponentsof v arenon-exteriorlocalcomponents.Theexteriorlocalcomponentscannotbelocalinnercomponentsotherwisewewouldnotbeabletoobtain anyintervalrepresentationofthebigraph.Hencethetwoexteriorcomponents C 1 ;C n mustbethetwosidecomponentsof v andtheremaining n )]TJ/F15 11.9552 Tf 11.426 0 Td [(2componentsarethe localinnercomponents.Thus B ' B 3 .ThisstructureisalsoillustratedinFigure 3.17. 46

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3.4Conclusionsandfuturework Weinvestigatedthestructuresandcharacteristicsofp-improperintervalbigraphs. Thesameideascouldbenaturallyextendedtointerval k -graphs.Thesearegraphs withapropercoloringwhereeachvertex v canbeassignedaninterval I v ofthereal linesuchthattwoverticesareadjacentifandonlyiftheircorrespondingintervals overlapandeachvertexhasadierentcolor.Itwouldbeinterestingtostudythe rolesplayedbyimproprietyandcriticalityoninterval k -graphs. 47

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4.Characterizationofunitprobeinterval2-trees 4.1Introduction IntervalgraphswereintroducedbyHajos[22],andwerethencharacterizedbythe absenceofinducedcyclesoflengthlargerthan3andasteroidaltriplesbyLekkerkerker andBoland[26]in1962.Unitintervalgraphsaretheintervalgraphsthathavean intervalrepresentationinwhicheachintervalhasunitlength.In1969,Roberts[36] provedthattheclassesofproperintervalgraphsandunitintervalgraphscoincideand heshowedthatintervalgraphsthathavenoinduced K 1 ; 3 areunitintervalgraphs. Muchmorerecentlyin2007Gardigaveamuchshorternewconstructiveproofof Robertsoriginalcharacterizationofunitintervalgraphs[18]. Agraphisaprobeintervalgraphifthereisapartitionof V G intosets P and N andacollection f I v : v 2 V G g ofintervalsofRsuchthat,for u;v 2 V G , uv 2 E G ifandonlyif I u I v 6 = andatleastoneof u or v belongsto P .Thesets P and N arecalledtheprobesandnonprobes,respectively.If,for G aprobeinterval graph,themembersof I v : v 2 V G areclosedintervalsofidenticallength,then G isaunitprobeintervalgraph.Theprobeintervalgraphmodelwasinventedin connectionwiththetaskcalledphysicalmappingusedinconnectionwiththehuman genomeprojectbyZhangandZhanget.al.[43],[44]. Onewaytodescribethestructureofaclassofgraphsisbyndingitscompletelist ofminimalforbiddeninducedsubgraphs;Nocompleteforbiddeninducedsubgraph characterizationforgeneralprobeintervalgraphshasbeenfoundsofar.Asthe characterizationofprobeintervalgraphsseemstobedicult,researchhasfocussed onclassesofprobeintervalgraphs. Familyof2-treesarethesetofallgraphsthatcanbeobtainedbythefollowing construction:ithe2-completegraph, K 2 ,isa2-tree;iitoa2-tree Q 0 with n )]TJ/F15 11.9552 Tf 12.021 0 Td [(1 vertices n> 2addanewvertexadjacenttoa2-completesubgraphof Q 0 .LiSheng rstcharacterizedcycle-freeprobeintervalgraphs[41].Asanaturalextensionof 48

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thecharacterizationfortrees,PrzuljandCorneilattemptedaforbiddensubgraph characterizationof2-treesthatareprobeintervalgraphsandtheyfoundatleast62 distinctminimalforbiddeninducedsubgraphsforprobeintervalgraphsthatare2trees[34].MorerecentlyBrown,FleschandLundgrenextendedthelistto69and gaveacharacterizationintermsofsparsespinyinterior2-lobsters[7].In2009Brown, ShengandLundgrengaveacharacterizationofcycle-freeunitprobeintervalgraphs [15].In[11]variouscharacterizationsaregivenfortheprobeintervalgraphsthatare bipartite.Thesecharacterizationsareforclassesofbipartitegraphs,butnogeneral characterizationofbipartiteprobeintervalgraphsweregiventhere.Howeverrecently, BrownandLangleycharacterizedunitprobeintervalgraphsforbipartitegraphs[10]. Thischapterrestrictstotheunitcaseofprobeintervalgraphswhichare2-trees. Inthischapterwecharacterize2-treesthatareunitprobeintervalgraphs.In Section2weintroducesomeimportantsubclassesof2-trees.Inaddition,weintroduce severalbasicresultsthatwillbeusefulinthecharacterization.IntheSection3we characterize2-caterpillarsandinterior2-caterpillarsintermsofforbiddeninduced subgraphsandshowthat2-treesthatareunitprobeintervalgraphshavetobeinterior 2-caterpillars.Whilethissignicantlyreducesourfocus,itturnsouttheproblem remainsverydicult.Amajorreasonforthisisthatevendeterminingwhich2pathsareunitprobeintervalgraphsisdicult.SoinSection4wegivealistof forbiddensubgraphsfor2-pathsthatareunitprobeintervalgraphs.TheninSection 5wecompletethecharacterizationfor2-pathswhichareunitprobeintervalgraphs. InSection6weextendthelistofforbiddensubgraphsfromsection4toincludethe remainingforbiddensubgraphsforinterior2-caterpillars.InSection7wecompletethe listofforbiddensubgraphsfor2-treeunitprobeintervalgraphsusing27subgraphs. Thesizeofthelistillustratesthedicultyofthisproblem. 4.2Preliminaries 49

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Figure4.1: Examplesofsome2-paths Inthissectionweintroducesomesubclassesof2-trees.Alsowegiveseveralbasic resultsthatwillbeusefulthroughoutthechapter.Todescribethestructureof2-trees, weusetheideaofa2-pathintroducedbyBeinekeandPippertin[2]. Denition4.2.1 [2]A 2 -pathisanalternatingsequenceofdistinct 2 and 3 -cliques, e o ;t 1 ;e 1 ;t 2 ;e 2 ;:::;t p ;e p ,startingandendingwitha 2 -cliqueandsuchthat t i contains exactlytwodistinct 2 -cliques e i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 and e i i p .The length of 2 -pathisthe number, p ,of 3 -cliques. Ingeneral2-pathsaremuchmorecomplexthanpaths.Aswewillseelaterthat characterizationfor2-pathswhichareunitprobeintervalgraphsisverydicult. Drawingtheconnectiontotreefurther,Proskurowskiintroducedthenotionofa 2-caterpillarin[32].A2-leafisavertexwhoseneighborhoodisa2-clique. Denition4.2.2 [32]A 2-caterpillar P isa 2 -treeinwhichthedeletionofall 2 -leaves resultsina 2 -path,calledthebodyof P .A 2 -caterpillar P isan interior2-caterpillar ifforany 2 -leaf v , v isadjacenttoallverticesofsome 2 -completesubgraph e i ofany longest 2 -pathof P . IntheSection3wewillobtainacharacterizationof2-caterpillarsandinterior 2-caterpillars. Denition4.2.3 The 2-distance betweena 2 -leafandalongest 2 -pathof G isthe lengthoftheshortest 2 -pathbetweenthem. Denition4.2.4 An asteroidaltriple ATin G isaset A ofthreeverticessuch 50

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Figure4.2: Ontheleftisa2-caterpillar,andontherightisaninterior2-caterpillar thatbetweenanytwoverticesin A thereisapaththatavoidsallneighborsofthe third. Itiswell-knownthatintervalgraphscannotcontainasteroidaltriples.However, probeintervalgraphscancontainasteroidaltriplesprovidedtherestrictionsinthe followinglemmasareobeyed.Thefollowinglemmaswillalsobeveryusefulinproving someoftheresultsinthepresentandforthcomingsectionsspeciallyinshowingthat certain2-treesthatareprobeintervalgraphsarenotunitprobeintervalgraphs. Lemma4.2.5 [41]AtleastonevertexinanATofaprobeintervalgraphmustbea nonprobe. Lemma4.2.6 [41]IneveryATtheremustexistanon-probevertexusuchthatthere existapathbetweentheothertwoverticesintheATthathasanon-probeinternal vertex. Lemma4.2.7 [6]Thevertexofdegree3inanyinduced K 1 ; 3 ofaunitprobeinterval graphmustbeaprobeandatleasttwooftheverticesofdegree1oftheinduced K 1 ; 3 mustbenon-probes. Denition4.2.8 A k -fan isa 2 -pathoflength k suchthatall e i 'sareincidentto acommonvertexwhichwecallthecenter;allotherverticesarecalledtheradial vertices. Denition4.2.9 4-fanisa2-treemadeupoffour2-cliquessuchthatallthecliques shareoneparticularvertexcalledthecentralvertexasinFigure4.3.Thevertices 51

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alongthecircumferenceofthe4-fanarecalledtheradialvertices.Radialverticeswith degree2arethe endradialvertices .Thevertexwithdegree3whichisequidistantfrom boththeendradialverticeswillbecalledthe centralvertex or midradialvertex . Denition4.2.10 The Merge oftwo2-trees T i and T j takesplaceiftheintersection oftheirvertexsetisnotnull.So V T i V T j 6 = where V T i and V T j denotes thevertexsetsof T i and T j respectively. Forconvenience,wewillintroducesomenotation.Fromnowontherightend pointandleftendpointoftheintervalcorrespondingtoavertexvwillbedenoted by r I v and l I v respectively.By u $ v wewillmeanadjacencybetween u and v . WewillalsousetheacronymUPIGforunitprobeintervalgraph. Wewillnamea4-fanbyalistofsixvertices,say a )]TJ/F19 11.9552 Tf 12.375 0 Td [(b )]TJ/F19 11.9552 Tf 12.374 0 Td [(c )]TJ/F19 11.9552 Tf 12.375 0 Td [(d )]TJ/F19 11.9552 Tf 12.374 0 Td [(e )]TJ/F19 11.9552 Tf 12.375 0 Td [(f with centerat c bywhichwemean a;b;d;e;f isthepathofradialverticesand c isthe centerofthe4-fan. Lemma4.2.11 Thecenterofa4-fanmustbeaprobe,themid-radialorthecentral vertexvertexandatleastoneoftheend-radialverticesmustbenon-probes;theother twoverticesmustbeprobes. Proof: Let F = f c;r 1 ;r 2 ;r 3 ;r 4 ;r 5 g ,Ebea4-fanwhere r 1 ;:::::;r 5 ,inorder,fromthe pathwhichcontainstheradialverticesand c isthecentervertex.ByBrown,Sheng, Lundgren, c isaprobeandatmostoneof f r 1 ;r 3 ;r 5 g isaprobeandhence r 2 and r 4 mustbeprobes.Nowobservetherepresentation I r 1 = I r 2 =[0 ; 1], I c = I r 3 =[1 ; 2], I r 4 = I r 5 =[2 ; 3],workswith r 1 , r 5 asnon-probes;if r 5 isaprobeor r 1 isaprobe, then I r 3 =[0 : 5 ; 1 : 5] I r 3 =[1 : 5 ; 2 : 5]canbeused. A3-sun,denotedF1anddepictedintheFigure4.6isa2-caterpillarformedby three3-cliqueswitha2-leafonthenon-interioredgeofthe3-cliqueatthemiddle.It 52

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Figure4.3: 4-fan Figure4.4: Unitprobeintervalrepresentationofa4-fan Figure4.5: Anillustrationwhereunitprobeintervalrepresentationofa4-fanfails towork 53

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Figure4.6: F1 isawell-knownfactthatthe3-sunisnotanintervalgraph.Itiseasytoshowthatit isaprobeintervalgraph.Inthenextlemmawewillprovethatitisnotaunitprobe intervalgraph. Lemma4.2.12 The2-treeF1-sunisnotaunitprobeintervalgraph. Proof: Letusassumethat F 1isaUPIG.ItcanbeobservedfromFigure4.7that F 1is aprobeintervalgraph.Withoutlossofgenerality,byLemma4.2.5,let u 6 beanon probe.Since u 2 and u 4 areadjacentto u 6 ,sotheyareprobes.ByLemma4.2.6, u 3 mustbeanonprobe.Hence u 1 and u 5 areprobes.Wenowattempttodrawtheunit proberepresentationofF1.Weknowthat I u 3 I u 1 6 = and I u 3 I u 5 6 = .Wemake theintervalsof u 1 and u 5 asfarapartaspossible.Thus I u 3 isankedonbothsides by I u 1 and I u 5 . I u 2 and I u 4 aredrawnsuchthat r I u 1
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Figure4.7: Probeintervalrepresentationofa3-sunorF1 intervalgraph. Wewillnowprovethatprobeintervalgraph E 1showningure4.8isnotaunit probeintervalgraph.Wewilllaterusethesetworesults F 1and E 1donothave unitprobeintervalrepresentationstocharacterizeinterior2-caterpillars. Lemma4.2.13 The2-treeprobeintervalgraphE1showninFigure4.8isnotaunit probeintervalgraph. Proof: Weknowthat E 1isaprobeintervalgraph.Letusassumethat E 1has aunitprobeintervalrepresentation.Vertices u 2 ;u 1 ;u 3 ;u 4 ;u 5 and u 6 forma4-fanas inFigure4.8.Thevertex u 2 isthecenterofthe4-fanandsoitmustbeaprobe.The vertex u 4 isthecentralradialvertexofthesame4-fanandsoitmustbeanon-probe. AsseeninFigurethat u 4 ;u 3 ;v 1 and u 5 formaninduced K 1 ; 3 withcenterat u 4 and so u 4 mustbeaprobewhichisacontradiction.Hence E 1isnotaunitprobeinterval graph. 4.3Characterizationof 2 -caterpillars 55

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Figure4.8: E1 Inthissectionwesignicantlyreducetheclassof2-treesweneedtoconsiderfor unitprobeintervalgraphs.Firstwereducetheclassof2-treeunitprobeinterval graphsto2-caterpillarsandthentointerior2-caterpillars.Thusinthissectionwe giveacompletecharacterizationof2-caterpillarsandinterior2-caterpillars.Wewill seethatthecompletelistofforbiddensubgraphsfor2-treeswhichare2-caterpillars consistsof4subgraphs.Furthermorewealsoprovethatall2-treeswhichareunit probeintervalgraphsmustbeinterior2-caterpillars.Thisresultsignicantlyreduces ourspectrumfromawiderangeofgraphstoarelativelysmallclassofinterior2caterpillars. Theorem4.3.1 A2-treeisa2-caterpillarifandonlyifitdoesnotcontainB1,B1', B2,B3asinducedsubgraphs. Proof: Let G bea2-treewhichisa2-caterpillar.Bythedenitionof2caterpillar, G isa2-treesuchthatafterdeletionofall2-leavesof G wegeta2-path, calledthebodyof P .If G = B 1 ;B 1 0 ;B 2 ;B 3then G f 2-leavesof G g isnota2-path asseeninFigure4.9andso G cannotcontainthemasinducedsubgraphs. Let G bea2-treewithout B 1 ;B 1 0 ;B 2 ;B 3asinducedsubgraphs.Suppose G isnot a2-caterpillar.Let e o ;t 1 ;e 1 ;t 2 ;e 2 ;:::;t p ;e p bealongest2-pathof G andlabelit P . 56

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Since G isnota2-caterpillar, G f 2-leavesof G g isnota2-path.Ifevery2-leafof G werea2-leafatdistance1from P then G f 2-leaves g wouldbea2-pathwhichis acontradiction.Hence G musthavea2-leafat2-distance2ormorefrom P andthis canhappenintwoways. G hasa2-pathoflengthatleast2originatingfromsome e i orithasa2-pathoflengthatleast2originatingfromapairofverticesofsome t i notequalto e i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 or e i Case:Assumethat G hasa2-pathoflengthatleast2originatingfromsome e i .Since P isassumedtobealongest2-pathof G ,thereexistsa2-pathoflengthat least2oneithersideof e i on P .LetXbea2-pathoflength2whichoriginatesfrom 2-clique e i .Thereforethesmalleststructuresatisfyingtheseconditionsisa2-path P oflength4withXstartingfromthemiddle2-clique, e i anditslengthis2.According toPrzuljandCorneil[34],thereare2non-isomorphic2-pathsoflength4whichare A 1 4 and A 2 4 asinFigure4.10.Therstvertex x 1 ofXhas2choicesofpositionsasin Figure4.11andthesecondvertex x 2 ofXhas2choicesagain.Thereforewegetfour 2-treesubgraphs A i , i =1 ; 2 ; 3 ; 4asinFigure4.11.Observingthat A i , i =1 ; 2 ; 3 ; areisomorphicto B 1and A 4 isisomorphicto B 1 0 ,itfollowsthat G cannotcontain these2-paths.Hence G cannothave2-pathsoflength2ormoreoriginatingfrom someinterioredge e i . Case:Nowletusassumethat G hasa2-pathXoflengthatleast2originating fromapairofverticesofsome t i notequalto e i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 or e i .Since P isassumedtobea longest2-pathof G ,thereexistsa2-pathoflengthatleast2oneithersideof t i on P .LetXbea2-pathoflength2whichoriginatesfromapairofverticesofsome t i notequalto e i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 or e i .Thereforethesmalleststructuresatisfyingtheseconditions isa2-path P oflength5withXstartingfromthemiddleclique, t i anditslength is3.AccordingtoPrzuljandCorneil[34],therearethreenon-isomorphic2-paths oflength5whichare A 1 5 ;A 2 5 ;A 3 5 giveninFigure4.12.Since t i hasjustoneedge otherthan e i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ;e i ,thereisexactlyonewaytoaddtherstvertex x 1 ofX.Let x 2 57

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Figure4.9: Forbiddensubgraphsfor2-caterpillar Figure4.10: Construction1 bethesecondvertexofX.Theverticesadjacentto x 2 are x 1 andoneofthevertices from N x 1 .Thuswegetsix2-trees A i ;i =1 ; 2 ; 3 ;:::; 6asinFigure4.13.Observe that A 1 ;A 2 ;A 3 ;A 4 and A 5 areisomorphicto B 3and A 6 isisomorphicto B 2.But G cannothave B 2 ;B 3asinducedsubgraphs.Hence G cannothaveany2-pathof lengthatleast2originatingfromapairofverticesofsome t i notequalto e i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 or e i . Hencefromcasesandwecanconcludethat G cannothaveany2-leafat 2-distance2ormore.So G mustbea2-caterpillar. 58

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Figure4.11: Construction2 Figure4.12: Construction3 59

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Figure4.13: Six2-treescalled A i 's Theorem4.3.2 A 2 -caterpillarisaninterior 2 -caterpillarifandonlyifitdoesnot containF1asaninducedsubgraph. Proof: Let G bea2-treewhichisa2-caterpillarwith[ e 0 ;t 1 ;:::::e n ],alongest 2-pathof G whichwecall P .Werstassumethatitisaninterior2-caterpillar.Soit cannothaveany2-leafadjacenttotheendpointsofanon-interioredge.Since F 1isa 2-caterpillarwithalongest2-pathbeing e 0 ;t 1 ;e 1 ;t 2 ;e 2 ;t 3 ;e 3 anda2-leaforiginating fromapairofverticesof t 2 notequalto e 1 or e 2 ,itisanon-interior2-caterpillar. Hence G cannothave F 1initasaninducedsubgraph. Nowweassume G isa2-caterpillarwithout F 1.If G isnotinteriorthenithas a2-pathoflength1originatingfromapairofverticesofsome t i 6 = e i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ;e i .Since P isassumedtobealongest2-pathof G ,thereexistsa2-pathoflengthatleast1on 60

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eithersideof t i on P .LetXbea2-pathoflength1whichoriginatesfromapairof verticesofsome t i notequalto e i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 or e i .Thereforethesmalleststructuresatisfying theseconditionsisa2-path P oflength3withXstartingfromthemiddleclique, t 2 anditslengthis1.AccordingtoPrzuljandCorneil[34],thereisjustone2-path oflength3whichis A 1 3 depictedinFigure4.10.Thusthevertex x 1 thatformsXis adjacenttoapairofverticesin t 2 suchthat x 1 = e 1 ;e 2 .Thus x 1 canbeaddedto P injustoneway,asinnumber3ofFigure4.11whichforms F 1.Thisisacontradiction since G doesnotcontainanysubgraphisomorphicto F 1.Hence G cannothavea 2-leafatanon-interioredge.So G mustbeaninterior2-caterpillar. Theorem4.3.3 A2treeUPIGGisaninterior 2 -caterpillar Proof: Let G be2-treeunitprobeintervalgraph.If G isnota2-caterpillar then,bytheorem4.3.1,itmustcontainoneof B 1 ;B 1 0 ;B 2 ; or B 3asaninduced subgraph.FurthermoreasseeninFigure4.9 B 1and B 1 0 contains E 1, B 2and B 3contains F 1asinducedsubgraphs.Hence G shouldalsocontain E 1orF1as inducedsubgraphsprovided G isnota2-caterpillarwhichisacontradictionsince E 1and F 1areforbiddensubgraphsforunitprobeintervalgraphs.Hence G cannot contain B 1 ;B 1 0 ;B 2or B 3asinducedsubgraphs.SobyTheorem4.3.1 G mustbea 2-caterpillar. Wehaveprovedthatany2-caterpillarisinteriorifandonlyifitdoesnotcontain F 1asaninducedsubgraph.But F 1isaforbiddensubgraphforunitprobeinterval graphs.Hence G cannotcontain F 1asaninducedsubgraphandsoitmustbean interior2-caterpillar. 4.4Forbiddensubgraphsfor2-pathswhichareunitprobeintervalgraphs Interior2-caterpillarscanbethoughtofas2-pathswith2-leavesonsomeofthe interioredges.Sowhencharacterizinginterior2-caterpillarswhichareunitprobe 61

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intervalgraphsthenaturalstepistocharacterize2-pathswhichareunitprobeinterval graphs.Oncewehaveacompletecharacterizationofthe2-pathunitprobeinterval graphswewillkeepadding2-leavesateverypossibleinterioredgetodeterminewhich structuresfailtobeaunitprobeintervalgraphsthusderivingalistofforbidden subgraphsforinterior2-caterpillarswhichareunitprobeintervalgraphs.Aswehave mentionedearlierthecharacterizationproblemfor2-pathsinitselfturnsouttobe verydicult.Inthissectionwegivealistofforbiddensubgraphsfor2-pathswhich areunitprobeintervalgraphs.Theforbiddensubgraphsarenamedas F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10and F 11.Allofthesesubgraphsareprobeintervalgraphs buttheyfailtohaveaunitprobeintervalrepresentation. ItisworthmentioningherethatweuseeitherLemma4.2.7orLemma4.2.11 forthefollowingproofs.WeuseLemma4.2.11fortheproofsthat F 2, F 3, F 4, F 4, F 5, F 6, F 10, F 11areforbiddensubgraphsforunitprobeintervalgraphsandLemma 4.2.7forprovingthat F 7, F 8, F 9arealsoforbiddensubgraphsforunitprobeinterval graphs. Lemma4.4.1 The2-pathF2-fangiveninFigure4.14isnotaunitprobeinterval graph. Proof: Letusassumethat F 2asshowninFigure4.14isaUPIG.Itcanbe observedfromthegurethat u 2 ;u 3 ;u 4 ;u 5 ;u 6 ;u 7 and u 2 ;u 1 ;u 3 ;u 4 ;u 5 ;u 6 arevertices oftwo4-fansandbothofthemhavecenterat u 2 .Thecentralradialvertexoftherst 4-fanis u 5 andthecentralradialvertexofthesecond4-fanis u 4 .Hencebylemma 4.2.11 u 4 and u 5 mustbenon-probes.But u 4 $ u 5 andhencewegetacontradiction. Thus F 2isnotaunitprobeintervalgraph. Lemma4.4.2 The2-path F 3 giveninFigure4.15isnotaunitprobeintervalgraph. Proof: Letusassumethat F 3isaUPIG.WecanseefromFigure4.15that x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 , x 1 , x 8 and x 10 arecentersof4-fansand x )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 , x 2 , x 7 and x 11 arecentralradialvertices 62

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Figure4.14: F2 Figure4.15: F3 ofthefour4-fans.Also x )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 and x 1 areendradialverticesofthe4-fanswithcenters at x 1 and x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 respectively.So x 1 and x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 mustbeprobes.Furthermore x 8 and x 10 arealsoendradialverticesofthe4-fanswithcentersat x 10 and x 8 respectively.So x 10 and x 8 mustbeprobesaswell.Hence x )]TJ/F17 7.9701 Tf 6.586 0 Td [(3 , x 3 , x 6 and x 12 theotherendradial verticesofthe4-fansmustbenon-probes.But x 3 $ x 6 andsowegetacontradiction. Lemma4.4.3 The2-path F 4 giveninFigure4.16isnotaunitprobeintervalgraph. 63

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Figure4.16: F4 Proof: Letusassumethat F 4isaUPIG.WecanseefromFigure4.16that x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 , x 1 , x 8 and x 11 arecentersof4-fansand x )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 , x 2 , x 9 and x 12 arecentralradial verticesofthefour4-fans.Also x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 and x 1 areendradialverticesofthe4-fanswith centersat x 1 and x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 respectively.Sotheymustbeprobes.Furthermore x 8 and x 11 arealsoendradialverticesofthe4-fanswithcentersat x 11 and x 8 respectively.So theymustbeprobestoo.Hence x )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 , x 3 , x 6 and x 13 theotherendradialverticesof the4-fansmustbenon-probes.But x 3 $ x 6 andsowegetacontradiction. Lemma4.4.4 The2-pathF5giveninFigure4.17isnotaunitprobeintervalgraph. Proof: Letusassumethat F 5isaUPIG.WecanseefromFigure4.17that x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 , x 1 , x 8 and x 12 arecentersof4-fansand x )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 , x 2 , x 9 and x 13 arecentralradialvertices ofthefour4-fans.Also x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 and x 1 areendradialverticesofthe4-fanswithcentersat x 1 and x )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 respectively.So x 1 and x )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 mustbeprobes.Furthermore x 8 and x 12 are alsoendradialverticesofthe4-fanswithcentersat x 12 and x 8 respectively.So x 12 64

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Figure4.17: F5 and x 8 mustbeprobestoo.Hence x )]TJ/F17 7.9701 Tf 6.586 0 Td [(3 , x 3 , x 6 and x 14 theotherendradialvertices ofthe4-fansmustbenon-probes.But x 3 $ x 6 andsowegetacontradiction. Lemma4.4.5 The2-pathF6giveninFigure4.18isnotaunitprobeintervalgraph. Proof: Letusassumethat F 6isaUPIG.WecanseefromFigure4.18 x )]TJ/F17 7.9701 Tf 6.586 0 Td [(2 , x 1 , x 9 and x 12 arecentersof4-fansand x )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 , x 2 , x 8 and x 13 arecentralradialvertices ofthefour4-fans.Hence x )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 and x 11 cannotbenon-probeswhicharealsotheend radialverticesforthesecondandthethird4-fan.So x 3 and x 6 theotherendradial verticesofthe4-fansmustbenon-probes.But x 3 $ x 6 andsowegetacontradiction. Lemma4.4.6 The2-pathF7giveninFigure4.19isnotaunitprobeintervalgraph. Proof: Letusconsidera2-treeprobeintervalgraph F 7asinFigure4.19. Assumethat F 7hasaunitprobeintervalrepresentation.Notethat u 4 isthecenter ofaninduced K 1 ; 3 formedby u 4 ;u 2 ;u 3 and u 6 .SobyLemma4.2.7 u 4 mustbea probe.Notethat u 6 isalsoacenterofaninduced K 1 ; 3 formedby u 6 ;u 4 ;u 7 and u 8 . So u 6 mustbeaprobeandatleasttwoof u 4 ;u 7 and u 8 mustbenon-probes.Since 65

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Figure4.18: F6 u 4 isaprobe, u 7 and u 8 mustbenon-probes.But u 8 isthecenterofanotherinduced K 1 ; 3 withleaves u 6 ;u 10 and u 12 andso u 8 mustbeaprobewhichisacontradiction. Lemma4.4.7 The2-pathF8giveninFigure4.20isnotaunitprobeintervalgraph. Proof: Letusconsidera2-treeprobeintervalgraph F 8whichisamergeof three4-fansasinFigure4.20withcentersat u 4 ;u 11 and u 6 .LetusassumethatF8 hasaunitprobeintervalrepresentation.Thevertex u 4 isthecenterofaninduced K 1 ; 3 formedby u 4 ;u 2 ;u 3 and u 6 so u 4 mustbeaprobe.Notethat u 6 isalsoacenter ofaninduced K 1 ; 3 withleaves u 4 ;u 7 and u 8 ;so u 6 mustbeaprobeandatleast twoof u 4 ;u 7 and u 8 mustbenon-probes.Since u 4 isalreadyaprobe,then u 7 and u 8 mustbenon-probes.Also u 9 and u 10 areadjacentto u 8 ,so u 9 and u 10 mustbe probes.Now, u 11 ;u 9 ;u 10 and u 13 formaninduced K 1 ; 3 withcenter u 11 .Soatleast twoverticesoutof u 9 ;u 10 and u 13 mustbenon-probes.But u 9 and ;u 10 areprobes, so u 13 istheonlyvertexwhichisanon-probeandhencewegetacontradiction. 66

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Figure4.19: F7 Lemma4.4.8 The2-pathF9giveninFigure4.21isnotaunitprobeintervalgraph. Proof: Letusconsidera2-treeprobeintervalgraph F 9whichisamergeof three4-fans G 1 ;G 2and G 3withcentersat x )]TJ/F17 7.9701 Tf 6.586 0 Td [(2 ;x 1 and x 6 asinFigure4.21.let usassumethat F 9hasaunitprobeintervalrepresentation. x )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 isthecenterofan induced K 1 ; 3 formedby x )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 ;x )]TJ/F17 7.9701 Tf 6.586 0 Td [(4 ;x )]TJ/F17 7.9701 Tf 6.586 0 Td [(3 and x 0 .So x )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 mustbeaprobeandeither x )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 Figure4.20: F8 67

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Figure4.21: F9 and x 0 or x )]TJ/F17 7.9701 Tf 6.586 0 Td [(3 and x )]TJ/F17 7.9701 Tf 6.587 0 Td [(4 arenon-probes.Again x 1 isthecenterofaninduced K 1 ; 3 formedby x 1 ;x )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ;x 2 and x 3 .So x 1 mustbeaprobeandeither x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 and x 2 or x 2 and x 3 arenon-probes. Case1:If x )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 and x 0 arenon-probes,then x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 mustbeaprobe.So x 2 and x 3 must benon-probesbyLemma4.2.7and4.2.11whichimpliesthattwoadjacentvertices x 0 and x 2 arebothnon-probeswhichisacontradiction.Hence x )]TJ/F17 7.9701 Tf 6.586 0 Td [(3 and x 0 cannotbe bothnon-probes. Case2:If x )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 and x )]TJ/F17 7.9701 Tf 6.586 0 Td [(4 arenon-probesthen x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 mustbeaprobe.So x 2 and x 3 must benon-probesbyLemmas4.2.7and4.2.11whichimpliesthat x 4 and x 5 areprobes. From G 3whosecenterisat x 6 wecanseethat x 6 ;x 4 ;x 5 and x 8 formaninduced K 1 ; 3 withcenter x 6 ,andsoatleasttwoof x 4 ;x 5 or x 8 mustbenon-probes.But x 4 and x 5 arealreadyprobesandsowegetacontradiction.HenceItisnotpossibleto constructaunitprobeintervalrepresentationof F 9. Lemma4.4.9 The2-path F 10 giveninFigure4.22isnotaunitprobeintervalgraph. 68

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Figure4.22: F10 Proof: Letusassumethat F 10isaUPIG.WecanseefromtheFigure4.22 that x )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 , x 1 , x 7 and x 12 arecentersof4-fans.Itcanalsobeobservedthat x )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 , x 2 , x 8 and x 11 arecentralradialverticesofthefour4-fans,andthusmustbenon-probes. Hence x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 x 0 , x 9 and x 10 mustbeprobes.Eachoftheseverticesisalsooneofthe endradialverticesofthe4-fans.Sotheotherendradialvertices x )]TJ/F17 7.9701 Tf 6.586 0 Td [(4 , x 3 , x 6 and x 13 ofthe4-fansmustbenon-probes.But x 3 $ x 6 andsowegetacontradiction. Lemma4.4.10 The2-pathF11giveninFigure4.23isnotaunitprobeinterval graph. Proof: Letusassumethat F 11isaUPIG.WecanseefromtheFigure4.23 that x )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 , x 1 , x 7 and x 10 arecentersof4-fanswhere x )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 , x 2 , x 8 and x 12 arecentral radialverticesofthefour4-fans.So x )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 , x 2 , x 8 and x 12 mustbenon-probes.Hence x )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 and x 0 mustbeprobesandtheyaretheendradialverticesof4-fanswithcenters at x 1 and x )]TJ/F17 7.9701 Tf 6.586 0 Td [(2 .Furthermore x 7 and x 10 arealsoendradialverticesofthe4-fanswith centersat x 10 and x 7 respectively.Since x 10 and x 7 arealsocenters,theymustbe probes.Hencetheotherendradialvertices, x )]TJ/F17 7.9701 Tf 6.587 0 Td [(4 , x 3 , x 6 and x 11 ofthe4-fans,must 69

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Figure4.23: F11 benon-probes.But x 3 $ x 6 andsowegetacontradiction. 4.52-pathunitprobeintervalgraphcharacterization Inthefollowingsectionwewillgiveacharacterizationof2-pathsthatareunit probeintervalgraphs.The2-pathcharacterizationturnedouttobequitedicult and,thecompletelistofforbiddensubgraphsforthischaracterizationincludes10 subgraphs.Inthebeginningofthesectionwewillgiveseveraldenitionswhichhave beenusedtoproceedthroughtheproofs.Wewillalsointroducesomeadditionaldefinitionsaswemovetowardsthenalgoal.Whileconstructinga2-pathwestartwith a2-completegraph, K 2 andcallit G .Nextwekeepaddingnewverticesadjacenttoa 2-completesubgraphof G suchthatthenewvertexisadjacenteithertothetwomost recentlyaddedverticesornot.Ifitisadjacenttothemostrecentlyaddedvertices wegeta2-pathwithno4-fanwhichwewillcallastraight2-pathwewillformally deneastraight2-pathlater.Ifnot,thenthenewvertex,afteraddition,either formsan F 1whichisnotallowedsinceitisnota2-pathoritcreatesa4-fan.Soat everystagewehaveexactlythreeplaceswherethenewvertexcanbeaddedprovided noinduced F 2 ::::::F 11isformedandasaresultweeithergetstraight2-paths, F 1s ormergeof4-fans.Henceitcanbeconcludedfromthediscussionsofarthatwith additionofavertextoanexisting2-patheithera4-faniscreatedornot.Thusthe 70

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structuresofa2-pathavoidingtheforbiddensubgraphsfrom F 2 ;::::::F 11,caneither beastraight2-pathormergeof4-fans.Atrstwegureoutthepossibleways4-fans canbemergedtoeachother. Denition4.5.1 Edge-consecutive4-fans:Let A i ;A i +1 betwo4-fans.The4-fans A i ;A i +1 willbecallededge-consecutiveif A i sharesatleastoneedgewith A i +1 , i=1,....n-1. Denition4.5.2 Vertex-consecutive4-fans:Let A i ;A i +1 betwo4-fans.The4-fans A i ;A i +1 willbecalledvertex-consecutiveif A i sharesexactlyonevertexwith A i +1 . Nowwewillprovesomeusefullemmasusingtheabovedenitionstondthe possibleshapesofa2-path.Firstwewilldealwith2-pathswhicharejustmergeof 4-fans.Asstatedearlier4-fansina2-pathcanbeeitheredge-consecutiveorvertexconsecutive.Firstwedeterminethepossiblestructuresofedge-consecutive4-fans. Lemma4.5.3 Twoedge-consecutive4-fanswithoutanF2-fancannotsharemore thantwo K 3 s. Proof: Let t 1 )]TJ/F19 11.9552 Tf 12.405 0 Td [(t 2 )]TJ/F19 11.9552 Tf 12.406 0 Td [(t 3 )]TJ/F19 11.9552 Tf 12.405 0 Td [(t 4 e 0 ;t 1 ;:::::t 4 ;e 4 bea4-fanA.letusaddanother 4-fanBtothissuchthat t 2 , t 3 and t 4 alsobelongstoB.Inotherwords t 2 , t 3 and t 4 aretheshared K 3 .Theonlywaywegeta4-fanfrom t 2 , t 3 and t 4 isbymakinga newvertexadjacentto e 4 asshowninpictureoftheFigure4.25.Itiseasytosee thatwegetanF2herewhichisacontradiction.Hencetwo4-fanscannotsharemore thantwo K 3 s. Lemma4.5.4 Twoedge-consecutive4-fanswithoutanF2-fancansharetwo K 3 s inexactlyonewayas A 4 6 intheFigure4.24. Proof: LetA1andA2betwoedge-consecutive4-fanswithoutanF2-fan suchthattheyshareexactlytwo K 3 s.Letuscallthisgraph G .SinceA1andA2 71

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Figure4.24: Non-isomorphic2-pathsoflength6 shareexactly2 K 3 s,then G mustcontainexactlysix K 3 s.Inotherwords G must bea2-pathoflength6.IthasbeenprovedinPrzuljandCorneilthattherearesix non-isomorphic2-pathsoflength6 A 1 6 ;A 2 6 ;A 3 6 ;A 4 6 ;A 5 6 ;A 6 6 asshowninFigure4.24.It canbeseeninthegurethat A 1 6 containsno4-fan,andthat A 5 6 ;A 6 6 haveexactlyone induced4-fan.Further A 2 6 ;A 3 6 haveaninducedF2inthem.Thus A 1 6 ;A 2 6 ;A 3 6 ;A 5 6 ;A 6 6 cannotbe G .Hence G mustbe A 4 6 Thefollowinglemmawilldeterminethepossiblestructuresanytwoedge-consecutive 4-fanswithoutanF2canhave. Lemma4.5.5 IfG1andG2aretwoedge-consecutive4-fansthatforma2-pathwith no F 2 ,thenG1andG2togetherareoneofthegraphsinFigure4.27. Proof: Let t 1 )]TJ/F19 11.9552 Tf 12.511 0 Td [(t 2 )]TJ/F19 11.9552 Tf 12.511 0 Td [(t 3 )]TJ/F19 11.9552 Tf 12.511 0 Td [(t 4 e 0 ;t 1 ;:::::;t 4 ;e 4 bea4-fan.Callit G 1 .Let G 2 beanother4-fan.Wecanmakethesetwo4-fansedge-consecutivewithoutforming aninduced F 2inthefollowingwayskeepinginmindthattheycannotsharemore thantwo3-cliques: Case1: G 1 and G 2 shareexactlytwo K 3 s: Thiscasehasbeendealtwithinthepreviouslemma.Itsconstructionisillustrated inthepictures2a,2b,2cofFigure4.25.2band2ccontainan F 1whichis nota2-pathandan F 2andsowedonotconsiderthem. Case2: G 1 and G 2 shareexactlyone K 3 : 72

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Figure4.25: Constructionthatshowspossiblemergeoftwo-4-fans 73

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Inthiscasewewillmerge G 1 = t 1 )]TJ/F19 11.9552 Tf 12.777 0 Td [(t 2 )]TJ/F19 11.9552 Tf 12.778 0 Td [(t 3 )]TJ/F19 11.9552 Tf 12.778 0 Td [(t 4 with G 2 suchthatthe3-clique t 4 becomestherst3-cliqueof G 2 .Since t 4 hastwoavailableedgestowhichthevertex from G 2 canbeadjacent,wegetexactly2waystoaddit.Oneofthewaysforman F 2asshowninpicture3cofFigure4.25andsowedonotconsideritanymore. Thusweconstruct t 4 )]TJ/F19 11.9552 Tf 12.333 0 Td [(t 0 3 of G 2 where t 0 3 isthesecond3-cliqueof G 2 .Nowatthis stagewearejustleftwith2optionsforthenextvertexfrom G 2 whichagaininturn hasjustoneoptionineachcaseforthenalvertexof G 2 .Thuswegetthestructures 3aand3boftheFigure4.25.3balsohasaninduced F 2andsowenolonger considerit. Case3: G 1 and G 2 shareexactlyoneedge: Inthiscasewewillmerge G 1 = t 1 )]TJ/F19 11.9552 Tf 11.136 0 Td [(t 2 )]TJ/F19 11.9552 Tf 11.136 0 Td [(t 3 )]TJ/F19 11.9552 Tf 11.136 0 Td [(t 4 with G 2 suchthattheedge e 4 becomes therstedgeof G 2 .Atthisstagewehaveexactlyonewaytoaddanewvertex. Afterthiswehaveexactlytwowaystoaddthenewvertex.Oneleadsto4a,4b andtheotherleadsto4c,4dofFigure4.25.Thisconstructionisillustratedin Figure4.26.Here4band4dhaveaninducedF2andsowedonotconsiderthem. 4cisanextensionof2a. Soournalstructuresaregivenby2a,3aand4aasillustratedinFigure4.25. Note:2aofFigure4.25isformediftwoedge-consecutive4-fansshareexactly two K 3 ,3aofFigure4.25isformediftwoedge-consecutive4-fansshareexactlyone K 3 and4aofFigure4.25isformediftwoedge-consecutive4-fansshareexactlyone edge. Lemma4.5.6 Ifagraphisamergeofthreegraphs G 1 ;G 2 ;G 3 whicharethreeedgeconsecutive4-fansthatforma2-pathwithno F 2 ;F 7 ;F 8 ;F 9 thenitisoneofthe followinggraphsinFigure4.31. 74

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Proof: Weconsider3caseshere: Case1: G 1 and G 2 shareexactlytwo K 3 sandG3shareseithertwo K 3 s,one K 3 oroneedgewith G 2 .Since2a,3aand4aasillustratedinFigure4.25arethe possiblestructureshere.Westartwith G 1 G 2 asthegraph2ainFigure4.25.So nowwehavetolookatthecasewhere G 2 canbeedgeconsecutivewith G 3 withno F 2.WeknowbyLemma4.5.5thatitcanbedoneinexactly3waysandhencewe get3structuresasshowninFigure4.28. Case2: G 1 and G 2 shareexactlyone K 3 and G 3 shareseithertwo K 3 s,one K 3 oroneedgewith G 2 .Since2a,3aand4aasillustratedinFigure4.25arethe possiblestructureshere.Westartwith G 1 G 2 asthegraph3aingure4.25.Sonow wehavetolookatthecasewhere G 2canbeedgeconsecutivewith G 3 withno F 2. Weknowbylemma4.5.5thatitcanbedoneinexactly3waysandhenceweget3 structuresasshownintheFigure4.29. Case3: G 1 and G 2 shareexactlyoneedgeandG3shareseithertwo K 3 s,one K 3 oroneedgewith G 2 .Since2a,3aand4aasillustratedinFigure4.25arethe possiblestructureshere.Westartwith G 1 G 2 asthegraph4aingure4.25.Asin thepreviouscases,nowwelookatthecasethat G 2 canbeedgeconsecutivewith G 3 withno F 2.AgainweknowbyLemma4.5.5thatitcanbedoneinexactly3ways andhenceweget3structuresasshowninFigure4.30. Since,fromFigure4.28areisomorphictofromFigure4.30,and4 fromFigure4.29respectivelyandfromFigure4.29isisomorphictofromgure 4.30,wegetsixnon-isomorphicstructureswhichareillustratedingure4.31three ofwhichareforbiddensubgraphs F 7, F 8, F 9. Wenowlookatthevertex-consecutivecase.Wealreadyknowthatedgeconsecutive4-fansarecapableofassumingexactly3structures.Thefollowingtwo lemmaswillprovethatvertex-consecutive4-fanscanassumeexactlyonestructure 75

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Figure4.26: Constructionoftwoedge-consecutive4-fans Figure4.27: Allpossibletwoedge-consecutive4-fansandtheirrepresentations 76

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Figure4.28: Formationofthreeedge-consecutive-4-fans Figure4.29: Formationofthreeedge-consecutive-4-fans 77

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Figure4.30: Formationofthreeedge-consecutive-4-fans Figure4.31: threeedge-consecutive-4-fans 78

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whichisalsoillustratedbelow. Lemma4.5.7 IfG1andG2aretwovertex-consecutive4-fansthatforma2-path withnoF2thenitisthegraphinFigure4.33. Proof: Let G bethemergeof G 1 and G 2 .Since G formsa2-path,itcannot have F 1asaninducedsubgraph.Itcanbeobservedthat G 1 and G 2 cannotshare anynon-endradialvertexsinceinthatcasethree K 3 sfromoneandone K 3 fromits complementwouldforman F 1.Hence G 1 and G 2 canshareonlytheendvertices. Henceif G 1 = u 1 )]TJ/F19 11.9552 Tf 11.855 0 Td [(u 2 )]TJ/F19 11.9552 Tf 11.856 0 Td [(u 3 )]TJ/F19 11.9552 Tf 11.855 0 Td [(u 4 )]TJ/F19 11.9552 Tf 11.856 0 Td [(u 5 )]TJ/F19 11.9552 Tf 11.856 0 Td [(u 6 withcenterat u 4 and G 2 = u 6 )]TJ/F19 11.9552 Tf 11.856 0 Td [(u 7 )]TJ/F19 11.9552 Tf 11.855 0 Td [(u 8 )]TJ/F19 11.9552 Tf -422.701 -23.98 Td [(u 9 )]TJ/F19 11.9552 Tf 10.816 0 Td [(u 10 )]TJ/F19 11.9552 Tf 10.817 0 Td [(u 11 withcenterat u 8 then u 6 isthesharedvertex.Thuswegetthepossible structureof G whichisillustratedinFigure4.33. Aconstructionof G from G 1 isalsoillustratedingure4.32. Lemma4.5.8 IfG1,G2,G3arevertex-consecutive4-fansthatforma2-pathwith noF2thenitisthegraphinFigure4.34. Proof: G 1 and G 2 shareexactlyonevertexand G 3 sharesonevertexwith G 2 . Sonowwehavetolookattheways G 2 canbevertex-consecutivewith G 3 withno F 2.Fromthepreviouslemmaweknowthatitcanbedoneinexactly1wayand hencewegetonlyonestructureasshowninFigure4.34. Nowwedenesomepossiblestructuresofa2-pathwhichhaveniceunitprobe intervalrepresentations.Therepresentationsarealsogivenbelow.Wewillalsouse theoperation+todenotemergebetween4-fans.Thesignplusisusedmostlyin Figuresforconvenience. Denition4.5.9 Straight2-path:Astraight 2 -pathisa 2 -pathwithno4-fansin thestructure.AnexampleisillustratedinFigure4.37. 79

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Figure4.32: Formationoftwovertex-consecutive-4-fans Figure4.33: Theonlystructurepossiblefortwovertex-consecutive-4-fans 80

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Figure4.34: Representationsofthreevertex-consecutive-4-fans Figure4.35: Anotherrepresentationofthreevertex-consecutive-4-fans 81

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Figure4.36: Threepossiblestructuresoftwoedge-consecutive4-fans Denition4.5.10 Staircase:A 2 -pathisastaircaseifitisamergeof4-fanswhere theconsecutive4-fansshareexactlytwo3-cliques.AnexampleisillustratedinFigure 4.36a. Denition4.5.11 2-snake:A 2 -pathisa 2 -snakeifitisamergeof4-fanswhere theconsecutive4-fansshareexactlytwoorone3-cliquesandthecentersofallthe 4-fansformapath.AnexampleisillustratedinFigure4.36b. Denition4.5.12 Extendedstaircase:A 2 -pathisanextendedstaircaseifitisa mergeof4-fanswheretheconsecutive4-fansshareexactlyoneedge.Thisisillustrated inFigure4.36c. Thefollowinglemmaisaveryusefulresultandhelpsalotinlaterproofswhere wegivetheunitprobeintervalrepresentationsofthe2-paths. Lemma4.5.13 The2-leavesonbothendsofa2-snakeandanextendedstaircase mustalwaysbenon-probes. Proof: Letusrstconsidera2-snakeasshowninFigure4.27.Here u 4 and u 6 arethecentersoftwo4-fans,sotheymustbeprobes.Again u 4 and u 6 arealsoone oftheendradialverticesofthe4-fanswithcenter u 6 and u 4 .Hencetheotherend radialverticesofthe4-fans u 2 and u 8 whicharealsotheend2-leavesofthe2-snake mustbenon-probes. LetusnowconsideranextendedstaircaseasshowninFigure4.27.Here x )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 82

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Figure4.37: Straight2-path and x 1 arethecentersoftwo4-fans.Sotheymustbeprobes.Thecentralradial vertices x )]TJ/F17 7.9701 Tf 6.586 0 Td [(3 and x 2 ofthetwo4-fansmustbenon-probes.Sotheiradjacentvertices x )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 and x 0 mustbeprobes.But x )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 and x 0 arealsooneoftheendradialvertices ofthe4-fanswithcentersat x 1 and x )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 respectively.Hencetheotherendradial vertices, x )]TJ/F17 7.9701 Tf 6.586 0 Td [(4 and x 3 ,whicharealsotheend2-leavesoftheextendedstaircasemust benon-probes. Lemma4.5.14 Astraight2-pathisaunitprobeintervalgraph. Proof: Itcanbeeasilyseenthatastraight2-pathdoesnothaveanyinduced K 1 ; 3 .Henceitisaunitintervalgraph.Thereforeitisaunitprobeintervalgraph. Lemma4.5.15 Astaircaseisaunitprobeintervalgraph. Proof: 83

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Figure4.38: Staircase Figure4.39: 2-snake 84

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Figure4.40: Extended-staircase+staircase Figure4.41: Staircaserepresentation 85

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Figure4.42: 2-snakerepresentation Figure4.43: extended-staircase+staircaserepresentation 86

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Let G beastaircaseasshowninFigure4.38. Let x 2 ;x 3 ;x 6 ;x 7 ;x 10 ;x 11 ;x 14 and x 15 benon-probes.Thusthenon-probesarevertices x i ;x i +1 whereistartsat i =2andiisbeingupdatedbyi:=i+3aftergetting incrementedby1ineveryiteration. ItiseasytoseefromtheFigure4.38thatifwestartcountingthe4-fansandthe edgesfromthebottomthen x 1 x 4 , x 5 x 8 , x 9 x 12 ..aretherstinterioredgesofthe rst4-fan,third4-fan,fth4-fan...respectively.Wewilldrawtheintervalsofthese verticessuchthat: I x 1 I x 4 = suchthattherightendof x 1 overlapswiththeleftendof x 4 . I x 5 I x 8 = suchthattherightendof x 5 overlapswiththeleftendof x 8 . I x 9 I x 12 = suchthattherightendof x 9 overlapswiththeleftendof x 12 . Thus I x i I x i +3 = suchthattherightendof x i overlapswiththeleftendof x i +3 . Startingat i =1,thevalueofigetsupdatedforeveryiterationby i = i +4.letus refertothefanverticesfrom x i to x i +3 for i =1 ; 5 ; 9 ;::: asverticesinboxes.Thus box k , k =1 ; 2 ; 3 ;::: correspondtovertices x i )]TJ/F19 11.9552 Tf 11.955 0 Td [(x i +3 for i =1 ; 5 ; 9 ;::: respectively. box k = f x 1+4 k )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 ;:::;x 4+4 k )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 g for box 1 ,let r I x 1
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Thuswehaveaunitprobeintervalrepresentationof G . Lemma4.5.16 A2-snakeisaunitprobeintervalgraph. Proof: Letusconsiderthe2-snakeinFigure4.39.Thecentralverticesofallthe4-fans involvedareprobes.Alltheverticesoftheedge-consecutive4-fansthatshareexactly one3-cliquearealsomadeprobes.Theunitprobeintervalrepresentationofa2-snake isgiveninFigure4.39. Lemma4.5.17 Anextendedstaircaseisaunitprobeintervalgraph. Proof: TheunitprobeintervalrepresentationofanextendedstaircaseisgiveninFigure 4.43. Itmustbenotedherethatmergeofextendedstaircaseswiththree4-fansdoes nothaveanyunitprobeintervalrepresentationsincetheyformaninduced F 9.It hasaunitprobeintervalrepresentationuntilthemergeoftwo4-fansbutoncethe third4-fanismergedwegetaninduced F 9.Bothanextendedstaircase-staircaseextendedstaircase-staircaseandanextendedstaircase-staircase-extendedstaircasestaircasehaveunitprobeintervalrepresentation,asshowningure4.40 Lemma4.5.18 Threeedge-consecutive4-fansthatforma2-pathhaveunitprobe intervalrepresentationifandonlyiftheydonothaveF2,F7,F8,F9asinduced subgraphs. Proof: Let G 1 , G 2 , G 3 bethreeedge-consecutive4-fansthatforma2-path.If theyhaveaunitprobeintervalrepresentationtheycannotcontain F 2 ;F 7 ;F 8 ;F 9as 88

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inducedsubgraphssincetheseareforbiddensubgraphsforunitprobeintervalgraphs. Fortheotherdirectionlet G 1 , G 2 , G 3 bethreeedge-consecutive4-fansthatform a2-pathwithno F 2 ;F 7 ;F 8 ;F 9asinducedsubgraphs.Bylemma4.5.6theyareone ofthegraphsinFigure4.31.Parts4,5,6ofFigure4.31formaninduced F 9 ;F 7 and F 8respectivelyandhencetheycannotbetakenintoconsideration.Sowehave toprovidetheunitprobeintervalrepresentationofparts1,2,3ofthesameFigure. Parts1and2areastaircaseanda2-snakewhoseunitprobeintervalrepresentation aregiveninFigure4.38andFigure4.39respectively,andtheunitprobeinterval representationofpart3oftheFigure4.31isgiveninFigure4.43. Lemma4.5.19 Anynumberofedge-consecutive4-fansthatforma2-pathhaveunit probeintervalrepresentationiftheydonothaveF2,F3,F4,F5,F6,F7,F8,F9, F10,F11asgeneratedsubgraphs. Proof: Let G 1 , G 2 , G 3 ,..... G n be n n> 2edge-consecutive4-fansthatform a2-pathwithno F 2 ;F 3 ;F 4 ;F 5 ;F 6 ;F 7 ;F 8 ;F 9 ;F 10 ;F 11asinducedsubgraphs. n =3:istruebytheabovelemma. n =4:Let G 1 , G 2 , G 3 , G 4 befouredgeconsecutive4-fans.Weknowthatthereare exactlythreeunitproberepresentablestructuresforthreeedge-consecutive4-fansas showninparts1,2,3oftheFigure4.31.Thusthemergeof G 1 , G 2 , G 3 mustbeone ofthosethreestructures. G 2 , G 3 , G 4 alsoformsoneofthem.Thuscorresponding toeachofthethreestructuresformedby G 1 , G 2 , G 3 wegetthreestructuresfrom G 2 , G 3 , G 4 .Thusintotalwehave10structures,fourofwhicharejustcontinuations of1,2and3ofFigure4.31.Since3isanon-symmetricstructureitscontinuation canbedonein2waysasshowninFigure4.40onebyrepeating3andtheother bydoing3andreverseof3consecutively.Wehaveaunitproberepresentationfor thestructuresthatarejustcontinuationsofthesametypeintheFigures4.38,4.39, 4.40forany n .Soweneedtofocusonstructureswhicharecombinationsofdierent 89

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kindsnamely1,2,3fromFigure4.31. Figures4.41,4.42,4.43givetheunitprobeintervalrepresentationforthree4-fans G 1 , G 2 , G 3 wherevertices u 1 ;u 2 ;u 3 ;u 4 ;u 5 ;u 6 areverticesformingtherst4-fanG1. Ifweaddthefourthedge-consecutive4-fanwegetthestructuresinFigures4.44, 4.45,4.46apartfromthestructuresdiscussedabove.Wewillconsiderthreecases correspondingtoFigures4.44,4.45,4.46. Case1:Figure4.44givesthepossiblestructuresobtainedfromFigure4.41. Case2:Figure4.45givesthepossiblestructuresobtainedfromFigure4.42. Case3:Figure4.46givesthepossiblestructuresobtainedfromFigure4.43. WelookatCase1rst.IfwecomparetherepresentationsinFigures4.41and4.44 itiseasytoseethattheintervalsoftheverticesof G 1 remainunchangedinboththe Figuresimplyingthattheadditionof G 4 doesnotaecttheintervalsofverticesfrom G 1 .Sotheremovalandsubsequentadditionof G 1 fromtherepresentationwillnot aecttheremainingrepresentationof G 2 , G 3 , G 4 .Furthermoreitisalsoeasytosee thattheadditionof G 1 impliesaddingatmost4verticeswhichisthecasewhen G 1 shareanedgewith G 2 andatleast2verticeswhen G 1 sharesexactly2 K 3 swith G 2 . Itcanbeseenfromthewaythesedierentrepresentationshavebeenlaidoutthat intervalsofverticesfrom G 1 G 2 thatgetaddeddonotoverlapwiththeintervalsof verticesfrom G 4 G 3 .Thesamecanbesaidabouttheothertwocases. n =5:Let G 1 , G 2 , G 3 , G 4 , G 5 beveedgeconsecutive4-fans.Wewillnowgeta representationof5veedge-consecutive4-fansinaniterativeway.Wewillrstdo therepresentationof G 3 , G 4 , G 5 .Sinceweknowthat G 2 canbeaddedto G 3 , G 4 and G 5 withoutaectingtheirrepresentationsuchthatthebiggergraph G 2 , G 3 , G 4 , G 5 hasaunitproberepresentation,werstaddtheintervalscorrespondingtothe remainingverticesof G 2 .Nowwejustconsider G 2 , G 3 , G 4 .Again,intervalsfrom G 1 canbeaddedtoitwithoutaectingtherepresentationof G 2 , G 3 , G 4 suchthat G 1 , G 2 , G 3 , G 4 hasaunitprobeintervalrepresentation.Soweaddtheremainingintervals 90

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Figure4.44: Fouredge-consecutive4-fans,structure1 correspondingtotheverticesof G 1 togettheunitprobeintervalrepresentationof G 1 , G 2 , G 3 , G 4 , G 5 .Alsotheintervalsfrom G 1 G 2 doesnotoverlapwiththeintervals from G 5 G 4 .Henceagainwecanconcludethatadditionof G 1 doesnotaectthe representationof G 2 , G 3 , G 4 , G 5 suchthat G 1 , G 2 , G 3 , G 4 , G 5 hasaunitprobe intervalrepresentation. Inthiswaywecankeepadding4-fanstothepresentstructuresandobtainaunit probeintervalrepresentationateverystep.Similarlywecanshowthatthisiterative processisvalidforthetheothertwocasestoo.Hencewecanconcludethatany numberofedge-consecutive4-fanshaveunitprobeintervalrepresentationiftheydo nothave F 2 ;F 7 ;F 8 ;F 9asinducedsubgraphs. Basedonthepreviousdiscussionswecanthinkofa2-pathasastraight2-path orbundlesof4-fansedgeorvertexconsecutiveortheirmerge.Sowewilllook ata2-pathasmergeofvertexoredgeconsecutive4-fansjoinedtogetherbystraight 2-paths.Thusa2-pathcanbeofthreetypes: 91

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Figure4.45: Fouredge-consecutive4-fans,structure2 Figure4.46: Fouredge-consecutive4-fans,structure3 92

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1.Groupsofedge-consecutive4-fansjoinedbystraight2-paths. 2.Groupsofvertex-consecutive4-fansjoinedbystraight2-paths. 3.AGroupofedge-consecutive4-fansjoinedbystraight2-pathwithagroupofedgeconsecutive4-fans. Thuswerstshowthateachofthegraphsobtainedfrom1,2,and3canhaveaunit probeintervalrepresentationiftheydonothave F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asgeneratedsubgraphs. Theorem4.5.20 Groupsofedge-consecutive4-fansjoinedbystraight2-pathshave unitprobeintervalrepresentationiftheydonothaveF2,F3,F4,F5,F6,F7,F8, F9,F10orF11asgeneratedsubgraphs. Proof: Let B 1 ;B 2 ;B 3 ;:::::B m be m m 2groupsofedge-consecutive4-fans thatforma2-pathwithno F 2 ;F 3 ;F 4 ;F 5 ;F 6 ;F 7 ;F 8 ;F 9 ;F 10 ;F 11asinducedsubgraphsandtheyarejoinedbystraight2-pathsoflength n .Letuscallthisstructure G .WewillnowgiveaunitprobeintervalrepresentationofG. Let B i and B j beanytwogroupsof4-fansjoinedbyastraight2-pathoflength n .Letuscallthis B ij .Sincetherepresentationofverticeslocatedononeendof B i isindependentoftherepresentationofverticesontheotherendof B i forany B i as seenfromFigure4.27and4.34.Sowecanconcludethatifthetheoremholdsfor B ij itshouldholdfortheentire G .Sowegiveaunitprobeintervalrepresentationof B ij fordierentvaluesof n . Case1 n =0:Thisisthecasewheretheintermittentstraight2-pathisoflength0.This impliesthatwehaveamergeof B i and B j suchthatthelastedgeof B i coincides withtherstedgeof B j .Both B i and B j aregroupsofedgeconsecutive4-fans.So theirmergeinthisfashiongivesalengthiersequenceofedge-consecutive4-fanswhose lengthisequaltothesumofthelengthsof B i and B j .Bythepreviouslemma B ij 93

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musthaveaunitprobeintervalrepresentation. Case2 n =1:Thisisthecasewheretheintermittentstraight2-pathisoflength1.Both B i and B j aregroupsofedgeconsecutive4-fans.Soweget2sequencesofedgeconsecutive4-fansjoinedbyastraight2-pathoflengthoneasshownincase n =1 inFigure4.47.Fromrepresentationsofthefouredge-consecutive4-fansasshownin Figures4.44,4.45,4.46itiseasytoseethatthelastaddedverticesofthe2-pathsare non-probesandtheirintervalscanbemadetohavedistinctrightendpoints.Here thelastvertexof B i iseithertherstvertexof B j ortheyareadjacentasshownin Figure4.48. Letusrstconsiderthecasewhenthelastvertexof B i istherstvertexof B j .If v n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 and v n arethelasttwoaddedverticesof B i thenintherepresentation r I v n )]TJ/F18 5.9776 Tf 5.757 0 Td [(1
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Both B i and B j aregroupsofedgeconsecutive4-fans.Lengthof B ij isequalto 2+length B i +length B j asshownincase n =2inFigure4.47.Weareinterested inthelasttwo4-fansof B i andthersttwo4-fansof B j .FromtherepresentationsinFigure4.27itiseasytoseethattheend2-leavesofextended-staircaseand 2-snakehavenon-proberestrictionsasshowninsecondandthirdpictureofFigure 4.27.Thustheonlystructurewhichhasnonon-proberestrictiononthe2-leafonone endisthestaircaseasshowninFigure4.27.Furthermorethe2-leafonthatendcan bemadetohaveadistinctendpoint.Thusonlywaytherecouldbeproblemsinthe representationif B i =2-snake,extendedstaircaseand B j =2-snake,extendedstaircase. AllpossiblestructuresobtainedinthiscasearegiveninFigures4.51,4.52,eachone ofwhichiseitheraforbiddensubgraphorhasaforbiddensubgraphasaninduced subgraph.Theonlypossiblestructurethatcanbeformedinthissituationisifright end2-leaffrom B i isnon-adjacenttotheleftend2-leafof B j asshownintheFigure 4.53.Sowelookatthecasewhenoneof B i or B j isastaircase.Withoutlossof generalitylet B i endinastaircase.Thedierentwaysinwhich B j canbeaddedtoit viaastraight2-pathoflength2aregiveninFigure4.54.Let v n bethe2-leafinthe rightendof B i and v n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 isthevertexaddedjustbefore v n . V n canbeeitherprobe ornon-probe.If v n isaprobethenbythesecondrepresentationofthestaircase, I v n hasadistinctrightendpointand r I v n )]TJ/F18 5.9776 Tf 5.756 0 Td [(1
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Figure4.54.Theonlynon-possibilityhereisthesubcaseinpicture1ofthesame gure.Thisisthecasewhen v n isadjacentto u 1 where u 1 istheleftend2-leafof B j and u 1 isapartofa2-snakeorextendedstaircase.Inotherwords u 1 isforcedtobe anon-probesothat v n isforcedtobeaprobeandso r I v n )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 3:Thisisthecasewheretheintermittentstraight2-pathisoflengthmorethan 3.Lengthof B ij isgreaterthan3+length B i +length B j asshownincase n =4 inFigure4.47.Inthiscasethetwoend2-leaves,infactthetwoend K 2 sarenot adjacentsincethereisatleastonevertexbetweenthem.Moreoveralltheverticesof thestraight2-pathalsohavenoprobe-non-proberestrictiononthem.Thuswecan easilydoaunitproberepresentation B ij bydoing B i rstandthencontinuingwith theintervalsofthestraight2-pathfollowedbytheintervalsofverticesfrom B j . Lemma4.5.21 Threevertex-consecutive4-fansthatforma2-pathhaveaunitprobe 96

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Figure4.47: Exampleofmergesfordierentvaluesof n Figure4.48: n =1:edge-consecutivemergefor2-paths 97

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Figure4.49: n =1:edge-consecutivemergefor2-paths intervalrepresentationifandonlyiftheydonothaveF2asageneratedsubgraph. Proof: Let G 1 , G 2 , G 3 bethreevertex-consecutive4-fans.Letitbeaunit probeintervalgraph.Thereforeitcannotconatain F 2asgeneratedsubgraph.Let G 1 , G 2 , G 3 bethreevertex-consecutive4-fansthatforma2-pathwithno F 2as inducedsubgraphs.SoitmustbethegraphinFigure4.34.Thesamepicturegives itsunitprobeintervalrepresentation. Note:wejustconsider F 2fortheabovelemmabecauseitistheonlystructure whichcroppedupduringtheconstructionofvertex-consecutive4-fansasseenearlier. Lemma4.5.22 Anynumberofvertex-consecutive4-fansthatforma2-pathwithout F2asageneratedsubgraphhaveaunitprobeintervalrepresentation. Proof: Let G 1 ;G 2 ;G 3 ;:::::G n be n n> 2vertex-consecutive4-fansthatform a2-pathwithno F 2asinducedsubgraphs.Weknowthateverysetof3vertex consecutive4-fans,say, A 1 = G 1 ;G 2 ;G 3 , A 2 = G 4 ;G 5 ;G 6 ,..., A k = G i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ;G i ;G i +1 ,....has aunitproberepresentationsimilartothatinFigure4.34.Sinceallthe A j 's j = 98

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Figure4.50: n =1:edge-consecutivemergefor2-paths 99

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Figure4.51: n =2:edge-consecutivemergefor2-paths 100

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Figure4.52: n =2:edge-consecutivemergefor2-paths Figure4.53: n =2:edge-consecutivemergefor2-paths 101

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Figure4.54: n =2:edge-consecutivemergefor2-paths 102

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Figure4.55: n =2:edge-consecutivemergefor2-paths Figure4.56: n =3:edge-consecutivemergefor2-paths 103

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Figure4.57: n =3:edge-consecutivemergefor2-paths Figure4.58: n =3:edge-consecutivemergefor2-paths 104

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1 ; 2 ;::::k;::: areagainvertexconsecutive,theyallsharetheendvertices.Thusthe rightend2-leafof A j istheleftend2-leafof A j +1 .Hencethelast2addedvertices of A j mustbeadjacenttotherst K 2 of A j +1 suchthatthelastandrstvertex coincides.Let v i , v i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 , v i )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 bethelast3addedverticesof A j .Thusif v i istheright end2-leaf A j then v i mustalsobetheleftend2-leafof A j +1 .Hence v i and v i +1 are theverticesoftherst K 2 of A j +1 .Let v i +2 bethirdaddedvertexof A j +1 .Itiseasy toseethat v i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 ;v i ;v i +1 formsa K 3 .Accordingtoarepresentation1inFigure4.34, v i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 and v i +1 areprobesand v i isnon-probe.Moreover r I v i )]TJ/F18 5.9776 Tf 5.756 0 Td [(2
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impliesthatwehaveamergeof B i and B j suchthatthelastedgeof B i coincides withtherstedgeof B j .Both B i and B j aregroupsofvertex-consecutive4-fans. Hencethelast2addedverticesof B i mustbetherst K 2 of B j .Let v n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 and v n be thelast2addedverticesof B i and u 1 , u 2 betherst K 2 of B j suchthat v n isthe rightend2-leafof B i and u 1 istheleftend2-leafof B j .Given B i thereare4waysin which B j canbeaddedtoitinthiscaseasshownintheFigure4.59.Thersttwo arethesubcaseswhentherightend2-leafof B i isadjacenttotheleftend2-leafof B j andthesecondtwoarethesubcaseswhenboththe2-leavescoincide.Pictures2 and4arenottakenintoconsiderationsincetheyformaninducedF2.Forpicture 1weusetherepresentationusedinthegure4.35.Thisisoksincerightend2-leaf of B i endwithdistinctrightendpoint,leftend2-leafof B j haveadistinctleftend pointandbothofthemareprobes.Forpicture3wemakeboththerightend2-leaf of B i andtheleftend2-leafof B j asnon-probes.Thiscanbedoneeasilysinceboth ofthemaresamehere.Thenweuserepresentation1oftheFigure4.34for B i and representation2oftheFigure4.34for B j . Case2 n =1:Thisisthecasewheretheintermittentstraight2-pathisoflength1.This impliesthatwehave B i and B j suchthatthelastvertexorrightend2-leafof B i coincideswiththeleftend2-leafof B j ortheyareadjacent.Both B i and B j are groupsofvertex-consecutive4-fans.Sointherstcaseweeithergetalengthier sequenceofvertex-consecutive4-fanswhoselengthisequaltothesumofthelengths of B i and B j plusoneoran F 2.Since F 2isavoidedandbythepreviouslemmawe knowthat B ij musthaveaunitprobeintervalrepresentation.Inthesecondcasetoo weeithergetanF2whichisavoidedorastructurethathasaunitprobeinterval representationasshownintheFigure4.60 Case3 n =2:Thisisthecasewheretheintermittentstraight2-pathisoflength2.This 106

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impliesthatwehave B i and B j suchthat B i and B j sharesnothingandtheright end2-leafof B i iseitheradjacenttotheleftend2-leafof B j oritisadjacenttothe othervertexoftherst K 2 of B j .Both B i and B j aregroupsofvertex-consecutive 4-fans.Soweget2sequencesofvertex-consecutive4-fansjoinedbyastraight2-path oflength2.Lengthof B ij isequalto2+length B i +length B j .Let v n betheright end2-leafof B i and u 1 betheleftend2-leafof B j suchthat v n )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 , v n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 and v n bethe last3addedverticesof B i and u 1 , u 2 , u 3 betherst3addedverticesof B j where u 1 , u 2 istherst K 2 of B j .Given B i ,thetotalpossiblewaysinwhich B j canbe addedto B i isillustratedintheFigure4.61andallofthemhaveunitprobeinterval representationsasshowninthesameFigure. Case4 n =3:Thisisthecasewheretheintermittentstraight2-pathisoflength3.This impliesthatwehave B i and B j suchthat B i and B j sharesnothingandwehave2 sequencesofvertex-consecutive4-fansjoinedbyastraight2-pathoflength3.Let v n bethe2-leafintherightendof B i and v n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 isthevertexaddedjustbefore v n .let u 1 bethe2-leafintheleftendof B j and u 2 isthevertexaddedjustafter u 1 .The onlynon-possibilityherecouldbethecasewhenboth v n and u 1 ,theboth2-leaves areadjacentwhichturnsouttobeoksince v n canbemadeaprobesuchthatinterval of v n hasarightdistinctendpointand u 1 anon-probesuchthatintervalof u 1 hasa leftdistinctendpointasillustratedintheFigure4.62. Case5 n> 3:Thisisthecasewheretheintermittentstraight2-pathisoflengthmorethan 3.Thisimpliesthatwehave B i and B j suchthat B i and B j sharesnothingand weget2sequencesofvertex-consecutive4-fansjoinedbyastraight2-pathoflength atleast4.Lengthof B ij isgreaterthan3+length B i +length B j .Inthiscasethe twoend2-leaves,infactthetwoend K 2 sarenotadjacentsincethereisatleastone vertexbetweenthem.Moreoveralltheverticesofthestraight2-pathalsohaveno 107

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Figure4.59: n =0:vertex-consecutivemergefor2-paths probe-non-proberestrictiononit.Thuswecaneasilydounitproberepresentation of B ij bydoing B i rstandthencontinuingwiththeintervalsofthestraight2-path followedbytheintervalsofverticesfrom B j . Theorem4.5.24 Agroupofvertex-consecutive4-fansjoinedtoagroupofedgeconsecutive4-fansviaastraight2-pathoflengthnn 0haveaunitprobe intervalrepresentationiftheydonothaveF2,F3,F4,F5,F6F7,F8,F9,F10,F11 asgeneratedsubgraphs. Proof: 108

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Figure4.60: n =1:vertex-consecutivemergefor2-paths 109

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Figure4.61: n =2:vertex-consecutivemergefor2-paths Figure4.62: n =3:vertex-consecutivemergefor2-paths 110

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Let G 1 and G 2 begroupsofvertex-consecutiveandedge-consecutive4-fansrespectively.Weknowthatanynumberofedge-consecutive4-fansthatforma2-path haveunitprobeintervalrepresentationiftheydonothave F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asinducedsubgraphs.Wealsoknowthatanynumberofgroupsof vertex-consecutive4-fanshaveunitprobeintervalrepresentationiftheydonothave F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asinducedsubgraphs.Let v n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 and v n bethelast2addedverticesof G 1 and u 1 , u 2 betherst K 2 of G 2 suchthat v n isthe rightend2-leafof G 1 and u 1 istheleftend2-leafof G 2 . Case1 n =0: Thelast2addedverticesof G 1 mustbetherst K 2 of G 2 . If v n isadjacentto u 1 thenwegetproblemintherepresentationfor G 2 =2-snake orextendedstair,sinceinthesecase v n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 = u 1 and u 1 isforcedtobeanon-probe from G 2 ,but v n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 isaprobefrom G 1 .Weget4structureshereallofwhichhave aninducedforbiddensubgraphinitasshowninFigure4.63.Thiscaseisokwith G 2 =staircaseandweavoidtheformationof5-fanssince u 1 canbemadeaprobe tooinastaircaseandweusetherepresentation3forvertexconsecutive4-fansto represent G 1 suchthat v n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 and v n areprobeand v n hasadistinctrightendpoint asshowninFigure4.64. If v n coincideswith u 1 thenwearegoodasshowninpicture5oftheFigure4.63and intheFigure4.64. Case2 n =1: Thereexistsexactlyone K 3 between G 1 and G 2 .Wehavetwocaseshere.Eitherthe 2-leavesareadjacentortheycoincidewhicharebothokasshowningure4.65. Case3 n =2: Thereexiststwo K 3 'sbetween G 1 and G 2 .ThepossiblestructuresandtheirrepresentationsaregiveninFigure4.66.Theonlynon-possibilityiswhen G 2 =2-snake, extendedstaircasewherethethe2-leavesarealwaysnon-probesandwegetforbidden 111

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Figure4.63: vertex-consecutive-edge-consecutivefor2-path subgraphs F 2 ;F 8 ;F 9asshowninFigure4.67. Case4 n 3: For n 3weusethesameargumentaswehaveusedforthevertexconsecutivecase anditworksne. ThefollowingLemmawillprovethatcertain2-pathsarealwaysunitprobeintervalgraphs.Afterthiswewillgeneralizeitfurtherandcontinuetoprovethatany 2-pathwillhaveaunitprobeintervalrepresentationifandonlyifitdoesnotcontain 112

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Figure4.64: vertex-consecutive-edge-consecutivefor2-path Figure4.65: n =1:vertex-consecutive-edge-consecutivefor2-path 113

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Figure4.66: n =2:vertex-consecutive-edge-consecutivefor2-path Figure4.67: n =2:vertex-consecutive-snake-edge-consecutivefor2-path 114

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F 2 ;F 3 ;F 4 ;F 5 ;F 6 ;F 7 ;F 8 ;F 9 ;F 10 ;F 11asaninducedsubgraph. Lemma4.5.25 Let G bea2-pathand G 0 = G )-394(f end 2 -leavesof G g .If G 0 isa straight2-path,2-snakeorstaircase,thenGisaunitprobeintervalgraph. Proof: Case1:Let G 0 beastraight2-path.Ifweaddbacktheend2-leaves thenwegettwopossible2-paths.Weeithergetacontinuationofthestraight2-path whichhasaunitprobeintervalrepresentationorastructureasshowninFigure 4.68.Theunitprobeintervalrepresentationisalsogiveninthesamegure. Case2:Let G 0 beastaircase.Ifweaddbacktheend2-leavesthenweeithergeta5fanoracontinuationofthestaircasewhichhasaunitprobeintervalrepresentation asshowninFigure4.68.Theunitprobeintervalrepresentationisalsogiveninthe samegure. Case3:Let G 0 bea2-snake.Ifweaddbacktheend2-leavesthenweeithergeta4-fan oracontinuationofthe2-snakeasshowninFigure4.70.Theunitprobeinterval representationofboththestructuresaregiveninthesamegure. Thefollowingtheoremwillgiveacompletecharacterizationof2-pathsthatare unitprobeintervalgraphs.Inthenextsectionswewilltaketheresultonestepfurther andgureoutthecompletelistofforbiddensubgraphsforinterior2-caterpillarswhich arejust2-pathswith2-leavesadjacenttosomeofinterioredges.Sinceinterior2caterpillarsbecomesa2-pathoncethe2-leavesareremoved,sothelistofforbidden subgraphsforinterior2-caterpillarswhichhaveunitprobeintervalrepresentation mustalsoincludethelistofforbiddensubgraphsfor2-pathswhichareunitprobe intervalgraphs. Theorem4.5.26 A2-pathisaunitprobeintervalgraphifandonlyifitdoesnot haveF2,F3,F4,F5,F6,F7,F8,F9,F10,F11asgeneratedsubgraphs. Proof: Let G beany2-path.Firstwewillprovethenecessarypart.Letus assumethat G isaunitprobeintervalgraph.Since F 2, F 3, F 4, F 5, F 6, F 7, F 8, 115

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Figure4.68: Straight-2-path+end-2-leaves 116

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Figure4.69: Staircase+end-2-leaves 117

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Figure4.70: 2-snake+end-2-leaves F 9, F 10, F 11donothaveunitprobeintervalrepresentations,anygraphwhichhasa unitprobeintervalintervalrepresentationcannotcontainthemasinducedsubgraphs. Thusany2-pathwithanunitprobeintervalintervalrepresentationcannotcontain themasinducedsubgraphs.So G doesnothave F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asgeneratedsubgraphs. Nowwewillprovethesucientpart.Letusassumethat G is2-pathwithout F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asinducedsubgraphs.Since G isa2-path itcanbeconsideredasgroupsofedgeorvertexconsecutive4-fansjoinedtogether bystraight2-paths.Since G doesnotcontainanyinduced F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11bytheprevioustheoremsitmusthaveaunitprobeinterval representation. Thusitcanbeconcludedthatthecompletelistofforbiddensubgraphsforany 2-pathisgivenby F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11. 118

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4.6forbiddensubgraphsforinterior 2 -caterpillarswhichareunitprobe intervalgraphs Inthissectionwewillprovidealistofinterior2-caterpillarsthatdonothaveany unitprobeintervalrepresentation.Wecallthem E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9 and E 10.Wehavealreadydiscussedtheinterior2-caterpillarE1insection2where wehaveprovedthatitdoeshaveanyunitprobeintervalrepresentation.Aswehave discussedearlier,2-pathscanbeconsideredastheskeletonofinterior2-caterpillars. Soifwearelookingatinterior2-caterpillarswhichhaveunitprobeintervalrepresentationwemustrstavoidthose2-pathsthatfailtobeunitprobeintervalgraphs. Fromtheprevioussectionwehavethecompletelistofforbiddensubgraphsof2-path unitprobeintervalgraphswhichare F 2 ;F 3 ;F 4 ;F 5 ;F 6 ;F 7 ;F 8 ;F 9 ;F 10and F 11. Furthermorewealsoknowthatsincewearenowdealingwithstructuresthatcome from2-pathsbyattaching2-leavestojusttheinterioredges,thenwemustavoid F 1 whichisnota2-pathtoo.Hencerightnow,toachieveourgoal,wemustprecisely lookatinterior2-caterpillarswhoseunderlying2-pathswhichitselfisF1-freedoes notcontain F 2 ;F 3 ;F 4 ;F 5 ;F 6 ;F 7 ;F 8 ;F 9 ;F 10 ;F 11asgeneratedsubgraphsandsee whichinterioredgesinthese2-pathsonbeingattachedto2-leavesposeaproblem forthemtocontinuetobeunitprobeintervalgraphs.Tondoutexactlywhich interioredgescannothave2-leavesattached,wewillrstloadthe2-pathunitprobe intervalgraphwith2-leavesoneveryedgeandtrytondthecorrespondingunit probeintervalrepresentations.Themomentwefailtogetonewegetourforbidden subgraphandhencethelistconsistingof Ei 'sisformed.Wealsohavesomeother importantresultsinthissectionmorepreciselythefollowingthreeresultswhich helpstolocatetheverticesinaninterior2-caterpillarwhichhaveprobe,non-probe restrictions. Lemma4.6.1 Ifthereis2-leafontherstorlast K 3 ofa4-fanthenitmustbea non-probealongwiththeendradialvertexthatislocatedonthe K 3 whichhasthe 119

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Figure4.71: Non-proberestrictionin4-fans 2-leaf. Proof: Letusconsiderthe4-fanwitha2-leaf v 1 ontheinterioredge u 3 u 6 of theend K 3 givenby u 3 u 6 u 5 asingure4.71.Thecentralradialvertex u 4 mustbea non-probe.So u 2 and u 6 mustbeprobes.Vertices u 3 ;u 5 ;v 1 ;u 2 formaninduced K 1 ; 3 withthecenterat u 3 .Since u 2 isaprobethen u 5 and v 1 mustbenon-probes. Lemma4.6.2 If G isastraight2-pathoflength n 4 ,denotedby e 0 ;t 1 ;::::::t n ;e n thathasa2leaf v 1 whichisadjacenttoan e i suchthat i 6 =0 ; 1 ;n )]TJ/F15 11.9552 Tf 10.772 0 Td [(1 ;n asinFigure 4.72then v 1 andtheverticeswhichareoppositetotheedge e i mustbenon-probes. Proof: Wewillprovetheresultbycontradiction.Withoutlossofgeneralitywe canassumethatn=4.Let G beastraight2-path e 0 ;t 1 ;::::::t 4 ;e 4 witha2-leafat e 2 asinFigure4.72andassumethat G isalsoaunitprobeintervalgraph.letusassume 120

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Figure4.72: Non-proberestriction thateither u 1 ;v 1 or u 4 isaprobe. Case1:Supposethat v 1 isaprobe.Itcanbeeasilyseenfromthegurethat u 2 ;u 0 ;v 1 ;u 4 formaninduced K 1 ; 3 withthecenterat u 2 .So u 2 mustbeaprobe. Since v 1 isaprobe, u 0 and u 4 mustbenon-probes.Furthermore, u 3 ;u 1 ;v 1 ;u 5 form anotherinduced K 1 ; 3 withcenterat u 3 .So u 3 mustbeaprobe.Similarly,since v 1 isaprobe, u 1 and u 5 mustbenon-probes.Hence u 0 and u 1 botharenon-probesbut u 0 $ u 1 andhencewegetacontradiction.So v 1 mustbeanon-probe. Theproofsforthecasesthateither u 1 or u 4 areprobesissimilar.Henceall3 verticesmustbenon-probes. Itiseasytoseethat G isaunitprobeintervalgraphwithrepresentationinFigure 4.73. Lemma4.6.3 Ifthereis2-leafonthemiddle K 3 ofa3-fanasshowninFigure4.74 then,itmustbeanon-probealongwiththeendradialvertexwhichislocatedonthe K 3 whichsharesthe2-leafwiththemiddle K 3 . Proof: Letusconsiderthe3-fanwitha2-leaf v 1 ontheinterioredge u 2 u 3 ofthemiddle K 3 givenby u 2 u 3 u 4 asshowninFigure4.74.Wewillprovethisby contradiction.Letusrstassumethat v 1 isaprobe.Itisobservablethat u 2 ;u 1 ;v 1 ;u 4 121

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Figure4.73: Unitprobeintervalrepresentation Figure4.74: 3-fanwitha2-leaf formsaninduced K 1 ; 3 .Since v 1 isaprobethen u 1 and u 4 mustbenon-probes. Furthermore, u 3 ;u 1 ;v 1 ;u 5 formsanotherinduced K 1 ; 3 .Since v 1 isaprobeso u 1 and u 5 mustbenon-probes.Henceboth u 4 and u 5 arenon-probes.But u 4 $ u 5 .Hence wegetacontradiction.Thus v 1 mustbeanon-probe. Nowweprovethateither u 1 ;v 1 ;u 5 or u 1 ;v 1 ;u 4 mustbenon-probes.Wealready knowthat v 1 isanon-probe.Letusassumethat u 1 isaprobewhichimplies u 4 ;u 5 mustbenon-probesbythesameargumentusedwhenweshowed v 1 anon-probe. Henceweagaingetthesamecontradiction.So u 1 mustalsobeanon-probe. Thenextseverallemmasestablishthelistofforbiddensubgraphs. Lemma4.6.4 The2-treeprobeintervalgraphwhichwewilldenotebyE2asin Figure4.75isnotaunitprobeintervalgraph. 122

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Figure4.75: E2 Figure4.76: probeintervalrepresentationforE2 Proof: Letusassumethat E 2hasaunitprobeintervalrepresentation.The vertices u 3 and u 6 areprobessincetheyarethecentralverticesof4-fans.Also u 4 and u 7 mustbenon-probessincetheyarethecentralradialverticesofthe4-fanswith centersat u 3 and u 6 .Hence u 2 ;u 5 and u 8 mustbeprobes.Again, u 3 ;u 2 ;v 2 ;u 6 isan induced K 1 ; 3 withthecenterat u 3 .Henceeither u 2 or u 6 mustbeanon-probe.But both u 2 and u 6 areprobeswhichisacontradiction.Hence E 2isnotaUPIG. Note1:FromtheabovetwoLemmasitcanbeconcludedthattwoedgeconnected 4-fanswhichrepresenta2-snakeandhaveunitprobeintervalrepresentationcanhave 2-leavesonlyintherstorlast K 3 ofthe2-pathotherwisewewillgetforbidden 123

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Figure4.77: E3 subgraphs E 1 ;E 2. Lemma4.6.5 The2-treeprobeintervalgraphwhichwewilldenotebyE3asin Figure4.77isnotaunitprobeintervalgraph. Proof: Letusassumethat E 3asinFigure4.77hasaunitprobeinterval representation.Vertices u 4 and u 7 areprobessincetheyarecentersoftwo4-fans. Also u 3 and u 8 mustbenon-probessincetheyarethecentralradialverticesofthe same4-fanswithcentersat u 4 and u 7 respectively.Hence u 1 ;u 5 ;u 6 ;u 10 mustbe probes.Itisobservablethat u 4 ;u 1 ;v 1 and u 6 isaninduced K 1 ; 3 withcenterat u 4 . Henceeither u 1 or u 6 mustbeanon-probe.Butboth u 1 and u 6 areprobeswhichis acontradiction.Hence E 3isnotaUPIG. Lemma4.6.6 The2-treeprobeintervalgraphwhichwewilldenotebyE4asin Figure4.78isnotaunitprobeintervalgraph. Proof: Letusassumethat E 4asinFigure4.78hasaunitprobeinterval representation.Vertices u 4 and u 7 areprobessincetheyarecentersoftwo4-fans. Alsovertices u 3 and u 8 mustbenon-probessincetheyarethecentralradialvertices 124

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Figure4.78: E4 ofthesame4-fanswithcentersat u 4 and u 7 respectively.Also u 5 ;u 4 ;v 1 and u 7 isaninduced K 1 ; 3 withcenterat u 5 .Henceeither u 4 or u 7 mustbeanon-probe. Butboth u 4 and u 7 arecentersof4-fansandhencetheymustbeprobeswhichisa contradiction.Hence E 4isnotaUnitprobeintervalgraph. Note2:Fromtheabovetwolemmasitcanbeconcludedthattwoedgeconnected 4-fanswhichrepresentanextended-staircaseandhaveunitprobeintervalrepresentationcanhave2-leavesonlyintherstorlast K 3 ofthe2-pathotherwisewewill getforbiddensubgraphs E 1, E 3, E 4. Lemma4.6.7 The2-treeprobeintervalgraphwhichwewilldenotebyE5asin Figure4.79isnotaunitprobeintervalgraph. Proof: Letusassumethat E 5hasaunitprobeintervalrepresentation.By Lemma4.6.2, u 1 ;v 1 and u 4 mustbenon-probes.Hence u 3 mustbeaprobe.Also u 5 ;u 3 ;v 2 and u 6 formaninduced K 1 ; 3 with u 5 asthecentervertex.Since u 3 isa probeSo v 2 and u 6 mustbenon-probes.Hence u 6 and u 4 arebothnon-probeseven though u 6 $ u 4 whichisacontradiction.So E 5cannothaveaunitprobeinterval representation. 125

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Figure4.79: E5 TheprobeintervalrepresentationofE5isgiveninFigure4.80. Lemma4.6.8 The2-treewhichwewilldenotebyE6asinFigure4.81isnotaunit probeintervalgraph. Proof: Letusassumethat E 6hasaunitprobeintervalrepresentation.By Lemma4.6.2 u 1 ;v 1 and u 4 arenon-probes.Thus v 2 mustbeaprobe.Since u 2 ;u 5 Figure4.80: ProbeintervalrepresentationofE5 126

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Figure4.81: E6 Figure4.82: ProbeintervalrepresentationofE6 and v 2 arethe3endsofa K 13 withcenterat u 3 ,then u 2 and u 5 mustbenon-probes. Since u 4 and u 5 areadjacentandbotharenon-probes,wegetacontradiction.Hence E 6isnotaUPIG. Lemma4.6.9 The2-treeprobeintervalgraphwhichwewilldenotebyE7asin Figure4.83isnotaunitprobeintervalgraph. Proof: Letusassumethat E 7asinFigure4.83hasaunitprobeinterval representation.Vertices u 4 and u 8 areprobessincetheyarecentersoftwo4-fans. Further u 6 ;u 4 ;v 1 ;u 8 isaninduced K 1 ; 3 withcenterat u 6 .Henceeither u 4 or u 8 or bothmustbeanon-probe.Butboth u 4 and u 8 arecentersof4-fansandhencethey mustbeprobeswhichisacontradiction.Hence E 7isnotaUPIG. 127

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Figure4.83: E7 Note3:FromtheaboveLemmaitcanbeconcludedthatiftwo4-fansarejoined byastraight2-pathoflengthonethenthatstraight2-pathcannothaveany2-leaf onanyofitsinterioredgeotherwisewewillgettheforbiddensubgraph E 7.Inother wordstwovertex-consecutive4-fansFan1andFan2cannothaveany2-leafonthe interioredgesofthe K 3 whichcontainsthecommonvertexsharedbybothFan1and Fan2. Lemma4.6.10 The2-treeprobeintervalgraphwhichwewilldenotebyE8asin Figure4.84isnotaunitprobeintervalgraph. Proof: Letusassumethat E 8asinFigure4.84hasaunitprobeinterval representation.Vertices u 3 and u 8 arecentersoftwo4-fanswith2-leaves v 1 and v 2 ontheirend K 3 's u 3 ;u 6 ;u 5 and u 7 ;u 8 ;u 9 respectively.SobyLemma4.6.1 u 5 and u 7 mustbenon-probes.But u 5 $ u 7 whichisacontradiction.Hence E 8isnotaUPIG. Lemma4.6.11 The2-treeprobeintervalgraphwhichwewilldenotebyE9asin Figure4.85isnotaunitprobeintervalgraph. Proof: Letusassumethat E 9asinFigure4.85hasaunitprobeinterval representation.Thevertices u 4 and u 10 arecentersoftwo4-fanswith2-leaves v 1 and 128

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Figure4.84: E8 Figure4.85: E9 v 2 ontheirend K 3 's u 4 ;u 6 ;u 5 and u 10 ;u 8 ;u 9 respectively.Soby4.6.1 u 6 and u 8 must benon-probes.But u 6 $ u 8 whichisacontradiction.Hence E 9isnotaUPIG. Lemma4.6.12 The2-treeprobeintervalgraphwhichwewilldenotebyE10asin Figure4.86isnotaunitprobeintervalgraph. Proof: Letusassumethat E 10asinFigure4.86hasaunitprobeinterval representation.Thevertices u 2 and u 9 arecentersoftwo4-fanswith2-leaves v 1 and v 2 ontheirend K 3 s u 2 ;u 6 ;u 5 and u 7 ;u 8 ;u 9 respectively.SobyLemma4.6.1 u 6 and u 7 mustbenon-probes.But u 6 $ u 7 whichisacontradiction.Hence E 10isnota UPIG. Lemma4.6.13 The2-treeswhichwewilldenotebyGroupE 11 asinFigure4.87is notaunitprobeintervalgraph. 129

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Figure4.86: E10 Proof: LetusassumethatthegraphsinFigure4.87haveaunitprobeinterval representation.Weknowthat u 1 isanon-probeand u 2 whichistheleftend2leafof2-snakeorextended-staircaseisalsoanon-probe.But u 1 $ u 2 whichisa contradiction. 1 4.7Characterizationofinterior 2 -caterpillarswhichareunitprobe intervalgraphs Thisisthenalsectionofourchapter.Inthissectionwecharacterize2-tree thatareunitprobeintervalgraphs.Insection3weprovedthat2-treeunitprobe intervalgraphsareinterior2-caterpillars.Ourmaingoalwastocharacterize2-tree unitprobeintervalgraphs.Sothisresultfromsection3reducedourrangeofgraphs drasticallyandwenarrowedourconcernstojustinterior2-caterpillarswhicharejust 2-caterpillarswithoutan F 1orinotherwordsa2-pathwith2-leavesonlyonthe interioredges.Firstwejustlookedat2-pathsandthenaddedthe2-leavestonally getthecharacterizationofinterior2-caterpillarwhichareunitprobeintervalgraphs. Usingthisandthestatedresultfromsection3weproveinthissectionthat2-trees areunitprobeintervalgraphsifandonlyiftheyareinterior2-caterpillarswithout Fi 'si=1,2,......11, Ej 'sj=1,2,....10andGroup-E11.Finallywehavetheproofofthe 130

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Figure4.87: Group-E11-u1,u2areend2-leavesof B i and B j 131

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Figure4.88: Possibleplacesof2-leaves characterizationof2-treeunitprobeintervalgraphs. FromNote1,Note2andNote3oftheprevioussectionwecanconcludethattwo edge-consecutiveorvertex-consecutive4-fanswhicharealsounitprobeintervalgraphs canpossiblyhave2-leavesonlyinthefollowingmannerasillustratedinFigure4.88. Theunitprobeintervalrepresentationofpictures1,2,3,4ofthesameFigure4.88 aregiveninFigures4.89,4.93,4.90,4.944.91,4.92.Soa2-snakewithallpossible 2-leaves,astaircasewithallpossible2-leavesandanextendedstaircasewithall possible2-leavesitmustbenotedherethatanextendedstaircasebeyondamerge oftwo4-fansgivesaforbiddensubgraphfora2-pathandsowedonotconsiderit, insteadwelookatitasamergewithastaircaseor2-snakehaveaunitprobeinterval representation. letusdiscusssomeimportantpointsabouta2-path.Wewillconstructa2-path fromthebeginning.Westartwitha2-completegraph, K 2 ,whichisa2-treeandcall it G .Wekeepaddingnewverticesadjacenttoa2-completesubgraphof G suchthat 132

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Figure4.89: Representationof2-snakewithallpossible2-leaves Figure4.90: Representationofextendedstaircasewithall2-leaves 133

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Figure4.91: Representationofstaircasewithall2-leaves Figure4.92: Representationofvertex-consecutive4-fanswithall2-leaves 134

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Figure4.93: Representationof2-snakewithall2-leaves Figure4.94: Representationofextended-staircaseandstaircasewithall2-leaves 135

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Figure4.95: Edge-consecutive4-fanswithall2-leaves Figure4.96: 4-fanwithall2-leaves 136

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thenewvertexiseitheradjacenttothetwomostrecentlyaddedverticesornot.Ifit isadjacenttothetwomostrecentlyaddedverticeswegeta2-treewhichisastraight 2-pathof2and3cliques.Wecallitastraight2-path. Ifnotthenweeithergetan F 1ora4-fan.Afteravoidingan F 1sinceweare buildinga2-pathifanewvertexisaddedanditisadjacenttoexactlyoneofthe2 mostrecentlyaddedvertices,thenitcreatesa4-fan. Henceitcanbeconcludedfromthediscussionsofarthatwiththeadditionof avertextoanexisting2-patheithera4-faniscreatedornot.Ifnotthenwegeta straight2-path.Wealsoknowthatwhena2-pathisamergeofmorethanone4-fan itcanbeeither:anedge-consecutiveoravertex-consecutive.Keepingthese inmind,2-pathscanbecategorizedinto3groupsinageneralizedwayasfollows: 1. Z 1 :2-pathswhichareformedbygroupsofedge-consecutive4-fansjoinedby straight2-paths. 2. Z 2 :2-pathswhichareformedbygroupsofvertex-consecutive4-fansjoinedby straight2-paths. 3. Z 3 :2-pathswhichareformedbyagroupofedge-consecutive4-fansjoinedby straight2-pathstoagroupofvertex-consecutive4-fans. Fromthesedenitionswederivethreecategoriesofinterior2-caterpillarsasfollows: 1 I 1 :Interior2-caterpillarswhichareformedbyattachingtheinterioredgesof Z 1 withallpossible2-leaves. 2 I 2 :Interior2-caterpillarswhichareformedbyattachingtheinterioredgesof Z 2 withallpossible2-leaves. 3 I 3 :Interior2-caterpillarswhichareformedbyattachingtheinterioredgesof Z 3 withallpossible2-leaves. Fromthepreviousresultsonthecharacterizationof2-pathsweknowthatthere exists3possiblestructuresfortwoedge-consecutive4-fansandonepossiblestructure fortwovertex-consecutive4-fans.Inthecaseofedge-consecutivetheyhavespecic 137

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namesasgiveninFigure4.27:astaircase b2-snake cextendedstaircase ThevertexconsecutivestructureisgivenbyFigure4.95.Wealsoknowthat boththe2-leavesofa2-snakeandextended-staircasehaveanon-proberestriction bylemma4.5.13.Furthermoreany2-leafonaninterioredgeoftherstorlast K 3 ofa4-fanmustalsobeanon-probe.Fromtheunitprobeintervalrepresentationofall thethreepossibletwoedge-consecutive4-fansandonepossibletwovertex-consecutive 4-fanswithallthepossible2-leaves,wecanconcludethattheintervalsof2-leaves canbeaddedtothemwithoutdisturbingtheintervallayoutoftheunderlying2-path suchthattheresultantgraphisalsoaUPIG.Furthermoreitisalsoworthnoting herethat E 1 ;E 2 ;E 3 ;E 4 ;E 7 ;E 8,GroupE 11aretheonlyforbiddensubgraphswhose underlying2-pathsareeitheredgeorvertexconsecutive4-fans.Therestarerelated tostraight2-paths.Wewillusethesefactstoprovethefollowinglemmas. Lemma4.7.1 Anynumberofedgeconsecutive4-fanswithallitspossible2-leaves thatformsa2-pathwithoutthe2-leaveshasaunitprobeintervalrepresentationif andonlyifthereisnoinducedF2,F3,F4,F5,F6,F7,F8,F9,F10andF11and E1,E2,E3,E4,E5,E6andE8init. Proof: Firstweprovethenecessarypart.Let G be n edgeconsecutive4-fans f 1 ;f 2 ;:::::::::f n withallpossible2-leaves.If G isunitaprobeinterval graphthen G cannotcontain F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11and E 1 ;E 2 ;E 3 ;E 4 ;E 5 ;E 6and E 8asinducedsubgraphs. Nowweprovethesucientpart.Let G be n edgeconsecutive4-fans f 1 ;f 2 ;:::::::::f n withallpossible2-leavesandnoinduced F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11, E 1, E 2, E 3, E 4, E 5, E 6and E 8init.Wewilltrytogetaunitprobeintervalrepresentationof G .Itcanbeobservedthatoutofalltheforbiddensubgraphs 138

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with2-leavesonly E 1 ;E 2 ;E 3 ;E 4 ;E 5 ;E 6and E 8canoccurinthecaseofedgeconsecutive4-fansandsoweavoidjustthesesubgraphsapartfromallthe Fi 'sto avoidanyproblemtowardstherepresentationof G .LetG'= G --leaves.SinceG' isa2-pathwithout F 2 ;F 3 ;F 4 ;F 5 ;F 6 ;F 7 ;F 8 ;F 9 ;F 10and F 11,soG'hasaunit probeintervalrepresentation.NowwelookatG'asmergeoftwoedge-consecutive 4-fans f 1 f 2 ;f 2 f 3 ;:::::f i f i +1 ;::::: .FromFigures4.89,4.90,4.91whichgivesdierent possiblemergeoftwoedge-consecutive4-fanswithallpossible2-leavessuchthat noEi'sareformedasinducedsubgraphsinit,itcanbeconcludedthattheintervalsof2-leavescanbeaddedtoeachoftheseblockswithoutdisturbingtheinterval layoutoftheunderlying2-pathsuchthattheresultantgraphisalsoaUPIGand the2-leavesdoesnothaveanydistinctendpoint.Sono2-leaf-intervalfromblock f i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 willoverlapwithintervalsfrom f i +1 .Moreoverallthe2-leavesarenon-probes sothattheiroverlappingwitheachothernevermatters.Wenowstartaddingthe intervalsof2-leavestotherepresentationofG'.Furthermore,since G cannothave E 1 ;E 2 ;E 3 ;E 4 ;E 7 ;E 8therearexedplaceswherethe2-leavesgetattachedtoG'. Henceaddingthe2-leavesof f 1 f 2 rstandinsertingthenewintervalscorresponding tothe2-leavesintherepresentationofG'doesnotgetaectedbythesuccessive additionof2-leafintervalsfor f 2 f 3 no2-leafintervalfrom f 3 overlapswithintervals from f 1 ,2-leafintervaladditionsalsodoesinitiateanychangeintherepresentationof theunderlying2-pathandoverlapping2-leavesdoesnotaecteachothersadjacency sincetheyareallnon-probes.Theadditionofintervalsof2-leavesisdone,keeping inmindthatno E 1 ;E 2 ;E 3 ;E 4 ;E 7 ;E 8isformedinthetransitionofG'to G ,tillthe wholegraphhasbeencovered.Thuswegetaunitprobeintervalrepresentationof G . Wenowlookattwovertexconsecutive4-fanswhichisaunitprobeintervalgraph fromtheprevioussectionwithallpossible2-leavesandgureouttheinterioredges wherewecanadd2-leaveswithoutdisturbingtheunitprobeintervalrepresentation 139

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ofthe2-path. Theorem4.7.2 Twovertex-consecutive4-fanswithallpossible2-leaveswithoutE1, E5,E6,E7hasaunitprobeintervalrepresentationasshowninFigure4.97. Proof: Let G betwovertex-consecutive4-fanswithmaximumpossible2-leavesasin Figure4.97.Wewillgiveaunitprobeintervalrepresentationof G . Weknowthatthemiddleedges u 7 u 8 and u 12 u 13 ofthe4-fanscannothave2-leaves otherwisewewillgetan E 1.Furthermoreonanypartofthebodyof G whichhasat leastsixconsecutive K 3 s,ifedge e k hasa2-leafthenitsnextnearest2-leafmustbeon theedge e k +3 otherwisewewillgettheforbiddensubgraphs E 5 ;E 6.Alsoitcannot haveany2-leafontheinterioredgeofthe K 3 thatmergesthetwo4-fansotherwise wewillgetaninduced E 7. Unitprobeintervalrepresentationof G : Letallthe2-leavesbenon-probes.Ifthe2-leaf v i isadjacenttotheedge e i then theverticesoppositetotheedge e i inthe K 0 3 s t i and t i +1 arealsonon-probes.We willconsidertheverticesfrom u 5 )]TJ/F19 11.9552 Tf 12.278 0 Td [(u 10 in G .Letuscallthis4-fan f i .Thus f i has u 5 ;u 6 ;u 7 ;u 8 ;u 9 ;u 10 ;v 3 and v 4 init.Wewillgivetheunitproberepresentationofthe verticesof f i andusingthethesamemethodwecangettheunitproberepresentations ofthenext4-fan u 10 )]TJ/F19 11.9552 Tf 10.18 0 Td [(u 15 called f j andnallywewillmergethesetworepresentations togettherepresentationof G .Wewillrstdrawtheintervalscorrespondingtothe non-2-leaveswhicharedenotedby u i sasinFigure4.97. In f i : r I u 5 140

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AlltheseareillustratedintheFigure4.97 Inthe f j : r I u 8
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Figure4.97: Vertexconsecutive4-fanswithall2-leaves 142

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Proof: Let G be n vertexconsecutive4-fans f 1 ;f 2 ;:::::::::f n withallpossible2-leaves.If G hasunitprobeintervalrepresentationthen G mustnotcontain F 2 ;E 1 ;E 5 ;E 6 ;E 7asinducedsubgraphsinit. Nowweprovetheotherdirection.Let G be n vertexconsecutive4-fans f 1 ;f 2 ;:::::::::f n withallpossible2-leaves. G doesnotcontainany F 2, E 1, E 5, E 6, E 7asinducedsubgraphsinit.Wewillprovethat G isaunitprobeintervalgraph. LetG'= G --leaves.G'isa2-pathmadeofvertex-consecutive4-fanswithout F 2 andsoG'hasaunitprobeintervalrepresentation.NowwelookatG'asmergeof twovertex-consecutive4-fans f 1 f 2 ;f 2 f 3 ;:::::f i f i +1 ;::::: .FromFigure4.97,itcanbe concludedthattheintervalsof2-leavescanbeaddedtoeachoftheseblockswithout disturbingtheintervallayoutoftheunderlying2-pathsuchthattheresultantgraph isalsoaUPIG.Itisalsoeasytoseethatoutofalltheforbiddensubgraphswith 2-leavesonly E 1 ;E 5 ;E 6 ;E 7canoccurinthecaseofvertex-consecutive4-fansandso weavoidjustthemapartfromthe F 2intheunderlyingstructure.Furthermore,itis alsoeasytoseefromthesamegurethatno2-leaffromanyinterioredgehasdistinct leftorrightendpointandsoitcanbeconcludedthattheinterior2-leaves,when addedtotheseblocks,doesnotoverlapwithanyintervaloutsidetheblock.Henceno 2-leafintervalfrom f j f j +1 overlapswithanyintervalfrom f j )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 and f j +1 .Also,wecan concludefromthesamegure,thattheremovalofintervalscorrespondingtothe2leavesfrominterioredgessuchthatweobtain2-pathswhicharevertex-consecutive 4-fansofanytwovertex-consecutive4-fans f i f i +1 doesnotaecttheunitprobe intervalrepresentationoftheunderlying2-pathoftheblock.Soafterthe2-leaves areaddedtoG',itcannotaecttherepresentationofany f i f i +1 .Sinceanynumber ofvertexconsecutive4-fanswhichforma2-pathisaunitprobeintervalgraph,we rstdotheunitproberepresentationofG'.Sincetheadditionof2-leavesonany twovertex-consecutive4-fans f i f i +1 isindependentoftheunitproberepresentation oftheunderlying2-pathandtherestoftheblocks,wecaneasilyaddtheintervals 143

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correspondingtotheinterior2-leavestotherepresentationofG'andnallygeta unitprobeintervalrepresentationof G . Wewillnowtakeeachof I 1 , I 2 , I 3 intoconsiderationsuchthatnowherethe forbiddensubgraphs E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 9, E 10 ; GroupE 11, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11areformedasinducedsubgraphsand giveaunitprobeintervalrepresentationforeachofthem. Theorem4.7.4 Interior 2 -caterpillarsgivenbystructuresisomorphictomergeof I 1 haveunitprobeintervalrepresentationifandonlyiftheydonothaveE1,E2,E3, E4,E5,E6,E7,E8,E9,E9,E10,F2,F3,F4,F5,F6,F7,F3,F8,F9,F10,F11as inducedsubgraphs. Proof: Let B 1 ;B 2 ;B 3 ;:::::B m be m m 2groupsofedge-consecutive4fanswithallpossible2-leavesandtheyarejoinedbystraight2-pathsoflength n n =0 ; 1 ; 2 :::: withallpossible2-leaves.Thisstructureisisomorphictomergeof I 1 andwewillcallit G .If G hasaunitprobeintervalrepresentationthenitmustnot contain E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asinducedsubgraphs. Nowweprovetheotherdirection.Let B 1 ;B 2 ;B 3 ;:::::B m be m m 2groupsof edge-consecutive4-fanswithallpossible2-leavesthatarejoinedbystraight2-pathsof length n n =0 ; 1 ; 2 :::: withallpossible2-leavessuchthatithasno E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asinduced subgraphs.Letusagaincallthisstructure G .Thus G isamergeof I 1 withoutthe forbiddensubgraphs.Wewillnowgiveaunitprobeintervalrepresentationof G for dierentvaluesof n . Let B i and B j beanytwogroupsofsuch4-fansjoinedbysuchastraight2-pathof length n .Letuscallthis B ij anditisisomorphicto I 1 .FromFigures4.96,4.89,4.91, 4.90itcanbeobservedthattherepresentationoftheverticesononeendofa4-fan 144

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oredge-consecutive4-fanswithall2-leavesdoesnotaectthevertexrepresentation ontheotherend.Sowecanusethesamestrategyforrepresentationonbothends andhenceifthetheoremholdsfor B ij itshouldholdfortheentire G .Henceitis sucienttogiveaunitprobeintervalrepresentationof B ij fordierentvaluesof n . Furthermore,since B i and B j haveallpossible2-leavesandtheyaremergeofedge consecutive4-fanssothelastandrst K 3 softheunderlying2-pathofboth B i and B j musthavea2-leafinitsinterioredge.Hencethelastaddedvertexontheright endof B i if B i isassumedtobeconstructedfromtheleftandtherstaddedvertex ontheleftendof B j mustallbenon-probes. Case1 n =0:Thisisthecasewheretheintermittentstraight2-pathisoflength0.This impliesthatwehaveamergeof B i and B j suchthatthelastedgeof B i coincides withtherstedgeof B j .Both B i and B j aregroupsofedgeconsecutive4-fanswith everypossible2-leave.Sowegetalengthiersequenceofsuchedge-consecutive4-fans whoselengthisequaltothesumofthelengthsof B i and B j .Bylemma4.7.1 B ij musthaveaunitprobeintervalrepresentation. Case2 n =1:Thisisthecasewheretheintermittentstraight2-pathisoflength1.This impliesthatwehaveamergeof B i and B j suchthat B i and B j sharesexactlyone vertex.Both B i and B j aregroupsofedgeconsecutive4-fanswithallpossible2leaves.Sowegettwosequencesofedge-consecutive4-fanssharingacommonvertex whoselengthisequalto1+length B i +length B j .Wehave2caseshere.Thelast addedvertexof B i iseithertherstaddedvertexof B j orthelastaddedvertexof B i isadjacenttotherstaddedvertexof B j . Letusrstconsiderthecasewhenthelastvertexof B i istherstvertexof B j .If u n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 and u n arethelasttwoaddedverticesof B i inorderand u n )]TJ/F17 7.9701 Tf 6.586 0 Td [(2 isthecenterofthelast 4-fanof B i thenbytherepresentationintheFigure4.96, r I u n )]TJ/F18 5.9776 Tf 5.756 0 Td [(2
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or r I u n )]TJ/F18 5.9776 Tf 5.756 0 Td [(2
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Case4 n =3:Thisisthecasewheretheintermittentstraight2-pathisoflength3.Both B i and B j aregroupsofedgeconsecutive4-fanswithallpossible2-leaves.Soweget 2sequencesofsuchedge-consecutive4-fans B i and B j joinedbyastraight2-pathof length3withallpossible2-leaves.Lengthof B ij isequalto3+length B i +length B j . Wewilllookatthelast4-fanof B i andtherst4-fanof B j withallpossible2-leaves init.Asdiscussedinthepreviouscase,since B i and B j haveallpossible2-leaves, thenthelastandrst K 3 softheunderlying2-pathofboth B i and B j musthavea 2-leafonitsinterioredge.Hencethelastaddedvertexontherightendof B i andthe rstaddedvertexontheleftendof B j mustallbenon-probes.Theonlypossible structuresthatcanformhereareillustratedinFigure4.101.Inothercasesweget adjacenciesbetweennon-probeverticeswhichgivesustheforbiddensubgraph E 9. TheunitproberepresentationsofthestructuresintheFigure4.101isgiveninthe samegure.Thesestructuresarepossiblebecausetherightend2-leaffrom B i or u n isnon-adjacenttotheleftend2-leafof B j or w 1 asshowninthesamegure. Furthermoreitcanbeeasilyseenfromthesestructuresthatwecannothaveany 2-leafontheinterioredgesoftheintermittent2-pathinpicture1ofthesamegure sincethatwillformeitheran E 1oran E 2,andwealsocanhaveexactlyone2-leaf ononeinterioredgeoftheintermittent2-patheitheron u n a or w 2 a asshownin picture2otherwisewewillgettheforbiddensubgraph E 6ifboththe2-leaveswere present,andwealsowillgeteitheran E 7or E 2ifthereisa2-leafon u n u n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 or w 1 w 2 respectively. Case5 n =4:Thisisthecasewheretheintermittentstraight2-pathisoflength4.Both B i and B j aregroupsofedgeconsecutive4-fanswithallpossible2-leaves.Soweget 2sequencesofsuchedge-consecutive4-fansjoinedbyastraight2-pathoflength4 withallpossible2-leaves.Thelengthof B ij isequalto4+length B i +length B j . 147

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Asbefore,wewilllookatthethelast4-fanof B i andtherst4-fanof B j withall possible2-leavesinit.Since B i and B j haveallpossible2-leaves,thenthelastand rst K 3 softheunderlying2-pathofboth B i and B j musthavea2-leafinitsinterior edge.Hencethelastaddedvertexontherightendof B i B i , B j areassumedto beconstructedfromtheleftandtheintervalsofverticesof B i , B j areplacedfrom theleftandtherstaddedvertexontheleftendof B j mustallbenon-probes. TheonlypossiblestructuresthatcanformhereareillustratedinFigure4.102.In othercasesweeithergetadjacenciesbetweennon-probeverticeswhichgivesusthe forbiddensubgraphs E 2 ;E 3 ;E 4 ;E 5 ;E 10orgetan F 2intheunderlying2-path.The unitproberepresentationsofthestructuresinFigure4.102isgiveninthesame gure.Therepresentationofthesestructuresarepossiblebecausetherightend2leaffrom B i isnon-adjacenttotheleftend2-leafof B j asshowninthesamegure. Furthermoreitcaneasilybeseenfromthesestructuresthatwecannothaveany2-leaf ontheinterioredgesoftheintermittent2-pathinpicture1ofthesameguresince thatwillformeitheran E 3oran E 4andwealsocanhaveexactlyone2-leafonone interioredgeoftheintermittent2-pathasshowninpicture2otherwisewewillgetthe forbiddensubgraph E 2 ;E 5 ;E 6 ;E 10.Theunitprobeintervalrepresentationof1and 2canbedonebyrepresenting B i rstasinrepresentationsofgure4.89,4.90,4.91 followedbytherepresentationofthestraight2-pathasinFigure4.103andnally representing B j againinthesamemanner.Picture3isisomorphicto2.Theunit probeintervalrepresentationofpicture4isgiveninthesamegure.Thisispossible sincetheend2-leavesofa2-snake,extended-staircaseandstaircasecanbemadeto havedistinctendpointswhichalsocanbeseenfromsamegures4.89,4.90,4.91.It canbenotedherethatPicture4istheonlydierentcaseherewhoserepresentation isgiveninthesamegure.Thiscaseisthestructurecontainingastraight2-pathof length10and3fromtheend4-fansand4fromtheintermittentstraight2-path oflength4. 148

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Furthermoreonanypartofthestructurewhichhasatleastsixconsecutive K 3 s,if edge e k hasa2-leafthenitsnearest2-leafmustbeontheedge e k +3 , e k )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 otherwise wewillgettheforbiddensubgraphsasshowninthepicture4ofthesameFigure. Case6 n> 4:Thisisthecasewheretheintermittentstraight2-pathisoflengthgreaterthan 4.Both B i and B j aregroupsofedgeconsecutive4-fanswithallpossible2-leaves. Soweget2sequencesofsuchedge-consecutive4-fansjoinedbyastraight2-pathof lengthgreaterthan4withallpossible2-leaves.Itcanbeconcludedfromtheprevious casestudythatifweconstructallthepossiblestructureswecangettwotypesof structuresasgivenbytheFigure4.102. Type1:Thesestructurescanberepresentedbydoing B i rstasintherepresentations ofthegure4.89,4.90,4.91followedbytherepresentationofthestraight2-pathas intheFigure4.103andnallyrepresenting B j againinthesamemanner. Type2:Structuresjoinedbystraight2-pathsoflengthsatleast11withallpossible 2-leavesatleast5fromtheintermittentstraight2-pathsand3eachfromthe4-fans ontwoends.The2-leavesontheinterioredgesofthestraight2-pathsarepresent sothatnoforbiddensubgraphsisformed.Soasstatedbefore,ifedge e k hasa2leafthenitsnearest2-leafmustbeontheedge e k +3 , e k )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 .Theunitprobeinterval representationofsuchstructuresarepossiblebecausetheend2-leavesofthe4-fans ontwoendswhicharepartsofthestraight2-pathcanbemadetohavedistinctend pointsandthentheintermittentstraight2-pathcanberepresentedasintheFigure 4.103. Furthermoreonanypartofthestructurewhichhasatleastsixconsecutive K 3 s,if edge e k hasa2-leafthenitsnearest2-leafmustbeontheedge e k +3 , e k )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 otherwise wewillgetforbiddensubgraphs. 149

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Figure4.98: n =1: I 1 Nowwelookatinterior2-caterpillarsisomorphicto I 2 whicharederivedfrom vertexconsecutive4-fansjoinedbystraight2-paths.Inthefollowingtheoremwegive aunitprobeintervalrepresentationofsuchinterior2-caterpillarswhentheydonot containanyoftheforbiddensubgraphsasinducedsubgraphsinit. Theorem4.7.5 Interior 2 -caterpillarsgivenbystructuresisomorphictomergeof I 2 haveunitprobeintervalrepresentationifandonlyiftheydonothaveE1,E2,E3, E4,E5,E6,E7,E8,E9,E9,E10,F2,F3,F4,F5,F6,F7,F3,F8,F9,F10,F11as inducedsubgraphs. Proof: Let B 1 ;B 2 ;B 3 ;:::::B m be m m 2groupsofvertex-consecutive4fanswithallpossible2-leavesthatarejoinedbystraight2-pathsoflength n withall possible2-leaves.Thisstructureisisomorphictomergeof I 2 andwewillcallit G .If G hasaunitprobeintervalrepresentationthenitmustnotcontain E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asinduced subgraphsbypreviousresults. Nowfortheotherdirectionlet B 1 ;B 2 ;B 3 ;:::::B m be m m 2groupsofvertex150

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Figure4.99: n =1: I 1 151

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Figure4.100: n =2: I 1 152

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Figure4.101: n =3: I 1 153

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Figure4.102: n =4: I 1 Figure4.103: Representationofstraight2-pathwithall2-leaves 154

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consecutive4-fanswithallpossible2-leavesthatarejoinedbystraight2-pathsof length n againwithallpossible2-leavessuchthatthereareno E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10or F 11asinduced subgraphs.Letusagaincallthisstructure G .Wewillnowgiveaunitprobeinterval representationof G fordierentvaluesof n . Let B i and B j beanytwogroupsofsuch4-fansjoinedbyasuchastraight2-path oflength n .Letuscallthis B ij ,anditisisomorphicto I 2 .Itiseasytoseethatif thetheoremholdsfor B ij itshouldholdfortheentire G sincerepresentationofthe verticesofoneendofany B i isnotdependentontherepresentationofverticesofthe otherendof B i forany i aswehaveseenearlier.Sowegiveaunitprobeinterval representationof B ij fordierentvaluesof n .Furthermore,since B i and B j haveall possible2-leavesandtheyaremergeofvertexconsecutive4-fanssothelastandrst K 3 softheunderlying2-pathofboth B i and B j musthavea2-leafonitsinterior edge.Hence,asintheprevioustheorem,thelastaddedvertexontherightendof B i andtherstaddedvertexontheleftendof B j mustallbenon-probes. Case1 n =0:Thisisthecasewheretheintermittentstraight2-pathisoflength0.This impliesthatwehave B i and B j suchthatthelastedgeof B i coincideswiththerst edgeof B j .Both B i and B j aregroupsofedgeconsecutive4-fanswithallpossible2leaves.Sowegettwopossiblecaseshere.Eithertherightend2-leafof B i isadjacent totheleftend2-leafof B j orthe2-leavescoincide.Ifthe2-leavesareadjacentthen wegettheforbiddensubgraph E 3asshownintheFigure4.104.Iftheycoincidethen weeithergetan F 2oraunitproberepresentationasshowninthesamegure. Case2 n =1:Thisisthecasewheretheintermittentstraight2-pathisoflength1.This impliesthatwehave B i and B j suchthat B i and B j sharesexactlyonevertex.Both B i and B j aregroupsofvertexconsecutive4-fanswithallpossible2-leaves.Sowe 155

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get2sequencesofvertex-consecutive4-fanssharingacommonvertexwhoselength isequalto1+length B i +length B j .Wehave2caseshere.Thelastaddedvertex of B i B i , B j isassumedtobeconstructedfromtheleftwithoutlossofgenerality iseithertherstaddedvertexof B j ortheyareadjacent. Letusrstconsiderthecasewhentherightendvertexof B i istheleftendvertexof B j .Thisisjustanothercaseofvertexconsecutive4-fanssince B i isvertexconsecutive to B j asseeninFigure4.105andsowehaveaunitproberepresentationofthis. Nowletusconsidertheothercasewheretheverticesareadjacent.Inthiscasewe gettheforbiddensubgraph F 2intheunderlying2-pathsoran E 2asshowninFigure 4.105. Case3 n =2:Thisisthecasewheretheintermittentstraight2-pathisoflength2.This impliesthatwehave B i and B j suchthat B i and B j sharesnothing.Both B i and B j aregroupsofvertex-consecutive4-fanswithallpossible2-leaves.Thelengthof B ij isequalto2+length B i +length B j .Wewilllookatthethelast4-fanof B i and therst4-fanof B j withallpossible2-leavesonit.Since B i and B j haveallpossible 2-leaves,thenthelastandrst K 3 softheunderlying2-pathofboth B i and B j must havea2-leafonitsinterioredge.Hencethelastaddedvertexontherightendof B i B i , B j isassumedtobeconstructedfromtheleftandtherstaddedvertexon theleftendof B j mustallbenon-probes.Thestructuresthatcanbeformedhere areillustratedintheFigure4.106.Iftherightend2-leafof B i isadjacenttotheleft end2-leafof B j thenwegetforbiddensubgraphs E 10or E 8asshowninthesame Figure.Intheothercasewhentherightend2-leaffrom B i or u n isnon-adjacent totheleftend2-leafof B j or w 1 wegetastructureasshowninFigure4.106.We dotheunitproberepresentationofthisstructureasillustratedinthesameFigure. Case4 n =3:Thisisthecasewheretheintermittentstraight2-pathisoflength3.This 156

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impliesthatwehave B i and B j suchthat B i and B j sharesnothing.Both B i and B j aregroupsofvertex-consecutive4-fanswithallpossible2-leaves.Thelengthof B ij is equalto3+length B i +length B j .Wewilllookatthethelast4-fanof B i andthe rst4-fanof B j withallpossible2-leaves.Since B i and B j haveallpossible2-leaves sothelastandrst K 3 softheunderlying2-pathofboth B i and B j musthavea2-leaf onitsinterioredge.Hencethelastaddedvertexontherightendof B i andtherst addedvertexontheleftendof B j mustallbenon-probeslikethepreviouscases.The onlypossiblestructuresthatcanhaveaunitprobeintervalrepresentationhereare illustratedinPictures1and2ofFigure4.101.Furthermorepicture1cannoteven haveany2-leafontheinterioredgesoftheintermittentstraight2-pathotherwise forbiddensubgraphs E 1or E 2areformedwhereaspicture2canhavejustone2-leaf inthestraight2-pathasshowninthesameFigure.Otherwiseforbiddensubgraph E 7 isformedifmorethanone2-leafisattachedtothestraight2-path.Intheothercase wegetadjacenciesbetweennon-probeverticeswhichgivesustheforbiddensubgraph E 9or F 2intheunderlying2-path. Case5 n =4:Thisisthecasewheretheintermittentstraight2-pathisoflengthequalto 4.Both B i and B j aregroupsofvertex-consecutive4-fanswithallpossible2-leaves. Soweget2sequencesofsuchvertex-consecutive4-fansjoinedbyastraight2-pathof lengthequalto4withallpossible2-leaves.The2-leavesontheinterioredgesofthe straight2-pathsarepresentsothatnoforbiddensubgraphisformed.Soasstated before,ifedge e k hasa2-leafthenitsnearest2-leafmustbeontheedge e k +3 , e k )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 .Let u n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ;u n bethelasttwoaddedverticesof B i suchthat u n istherightend2-leafof B i . Weknowthat u n mustbenon-probeand u n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 mustbeaprobe.Furthermoreeither u n or u n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 canbemadetohaveadistinctrightendpointasrequired.Thesamecan beconcludedabouttheleftend2-leafof B j .Let w 1 ;w 2 bethersttwoaddedvertices of B j suchthat w 1 istheleftend2-leafof B j .Weknowthat w 1 mustbenon-probe 157

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and w 2 mustbeaprobe.Furthermoreeither w 1 or w 2 canbemadetohaveadistinct leftendpointasrequired.Also,sincetheintermittentstraight2-pathisoflength atleast4,thereexistsatleastonevertexfromthe2-pathbetween u n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ;u n and w 1 ;w 2 suchthattheend2-leavesfrom B i and B j areneveradjacent.Welookathowthelast 4-fanfrom B i connectstotherst4-fanfrom B j viathestraight2-pathoflength4. ThedierentpossiblestructuresareshowninFigure4.102.Furthermoreonanypart ofthestructurewhichhasatleastsixconsecutive K 3 s,ifedge e k hasa2-leafthen itsnearest2-leafmustbeontheedge e k +3 , e k )]TJ/F17 7.9701 Tf 6.586 0 Td [(3 otherwisewewillgettheforbidden subgraphs.Theunitprobeintervalrepresentationsof1and2aredonebydoing B i rstasingure4.97followedbytherepresentationofthestraight2-pathasinFigure 4.103andnallyrepresenting B j againasingure4.97.ThegraphinPicture3is isomorphictothegraphinpicture2.Theunitprobeintervalrepresentationofthe graphinpicture4isgiveninthesamegure.Inall3representationsweuseoneof the2representationsfromFigure4.97for B i and B j . Case6 n> 4:Thisisthecasewheretheintermittentstraight2-pathisoflengthgreaterthan 4.Thisimpliesthatwehave B i and B j suchthat B i and B j sharesnovertex.Both B i and B j aregroupsofvertex-consecutive4-fanswithallpossible2-leaves.Sowe get2sequencesofsuchvertex-consecutive4-fansjoinedbyastraight2-pathoflength greaterthan4withallpossible2-leaves.Itcanbeconcludedfromthepreviouscase studythatifweconstructallthepossiblestructureswewillnevergetadjacencies betweentheend2-leavesof B i and B j andtheywouldhaveatleast2intermittent verticesbetweenthem.The2-leavesontheinterioredgesofthestraight2-pathsare presentsothatnoforbiddensubgraphisformed.Soasstatedbefore,ifedge e k hasa 2-leafthenitsnearest2-leafmustbeontheedge e k +3 , e k )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 .Let u n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ;u n bethelast twoaddedverticesof B i suchthat u n istherightend2-leafof B i .Weknowthat u n mustbeanon-probeand u n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 mustbeaprobe.Furthermoreeither u n or u n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 can 158

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Figure4.104: n =0: I 2 bemadetohaveadistinctrightendpointasrequired.Thesamecanbeconcluded abouttheleftend2-leafof B j .Let w 1 ;w 2 bethersttwoaddedverticesof B j such that w 1 istheleftend2-leafof B j .Weknowthat w 1 mustbenon-probeand w 2 must beaprobe.Furthermoreeither w 1 or w 2 canbemadetohaveadistinctleftendpoint asrequired.Also,sincetheintermittentstraight2-pathisoflengthatleast4there existsatleasttwoverticesfromthe2-pathbetween u n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ;u n and w 1 ;w 2 suchthat theend2-leavesfrom B i and B j areneveradjacent.Likethepreviouscase,theunit probeintervalrepresentationofsuchstructuresaredonebyrepresenting B i rstas ingure4.97thentheintermittentstraight2-pathcanberepresentedasinFigure 4.103followedby B j asingure4.97. Nowwelookatinterior2-caterpillarsisomorphicto I 3 whicharederivedfrom vertexconsecutive4-fansjoinedbystraight2-pathswithedge-consecutive4-fans. Inthefollowingtheoremwegiveunitprobeintervalrepresentationsofsuchinterior2-caterpillarswhentheydonotcontaincertainforbiddensubgraphsasinduced subgraphs. Theorem4.7.6 Interior 2 -caterpillarsgivenbystructuresisomorphictomergeof I 3 haveunitprobeintervalrepresentationifandonlyiftheydonothaveE1,E2,E3, 159

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Figure4.105: n =1: I 2 Figure4.106: n =2: I 2 160

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Figure4.107: n =3: I 2 161

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E4,E5,E6,E7,E8,E9,E9,E10,F2,F3,F4,F5,F6,F7,F3,F8,F9,F10,F11as inducedsubgraphs. Proof: Let G 1 and G 2 begroupsofvertex-consecutiveandedge-consecutive4-fansrespectivelyjoinedbyastraight2-pathoflength n ,withallpossible2-leaves.Letus callthisgraph G .Thisstructureisisomorphicto I 3 .If G hasaunitprobeinterval representationthenitmustnotcontain E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asinducedsubgraphs. Nowfortheotherdirectionlet G 1 and G 2 begroupsofvertex-consecutiveand edge-consecutive4-fansrespectivelyjoinedbyastraight2-pathoflength n ,withall possible2-leaves.Letuscallthisgraph G .Hence G isisomorphicto I 3 . G doesnot contain E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asinducedsubgraphs.Wewillgiveaunitprobeintervalrepresentation of G .Weknowthatanynumberofedge-consecutive4-fanswithallpossible2-leaves thatforma2-pathwithoutthe2-leaveshaveunitprobeintervalrepresentation iftheydonothave E 1, E 2, E 3, E 4, E 5, E 6, E 8, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asinducedsubgraphs.Wealsoknowthatanynumberofgroupsof vertex-consecutive4-fanswithallpossible2-leavesthatformsa2-pathwithoutthe 2-leaveshaveunitprobeintervalrepresentationiftheydonothaveF2,E1,E5,E6, E7asinducedsubgraphs.Let v n )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 and v n bethelast2addedverticesof G 1and u 1 , u 2 betherst K 2 of G 2.Thus v n istherightend2-leafoftheunderlying2-pathof G 1and u 1 istheleftend2-leafoftheunderlyingpathof G 2 Case1 n =0:Thelast2addedverticesof G 1 mustbetherst K 2 of G 2. Sinceweconsider G 1 and G 2 withallpossible2-leavesinthem,thelastandrst K 3 of G 1and G 2respectivelymusthave2-leavesonitsinterioredges.Henceboth v n and u 1 mustbenon-probes.Henceif v n isadjacentto u 1 thenwegetaprobleminthe representationof G aswegettheforbiddensubgraph E 3asaninducedsubgraphinit 162

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asshowninFigure4.108.Otherwise v n = u 1 andwegetaunitproberepresentation asshowninFigure4.108 Case2 n =1:Ifthereexistsexactlyone K 3 between G 1 and G 2 thenwehaveto considertwosubcases.Eitherthe2-leavescoincidewhichisokasshowninthegure 4.98ortheyareadjacentwhichformsaninduced E 2asshowninFigure4.109. Case3 n =2:Ifthereexiststwo K 3 between G 1 and G 2 thenwegetthefollowing subcases.Thepossiblerepresentablestructuresandtheirrepresentationsaregiven intheFigure4.100.Thenon-possiblityoccurswhenthenon-probe2-leavesof G 1 and G 2areadjacentandwegetforbiddensubgraphs F 2 ;E 8 ;E 10asshowninFigure 4.110. Case4 n =3:Ifthereexiststhree K 3 between G 1 and G 2 thenwegetthefollowing subcases.Thepossiblerepresentablestructuresandtheirrepresentationsaregiven inFigure4.101Since G 1 isvertex-consecutiveand G 2 isedge-consecutive,weuse representation2ofFigure4.97for G 1 forpicture1andrepresentation1ofFigure 4.97for G 1 forthepicture2.FurthermoreweuserepresentationsforFigures4.89, 4.90,4.91for G 2 inbothpictures1and2dependingonthekindof G 2 thatispresent there.Similartothepreviouscasesthenon-possibilityoccurswhenthenon-probe 2-leavesof G 1 and G 2 areadjacentandwegetforbiddensubgraphs F 2or E 9. Case5 n =4:Ifthereexists4 K 3 between G 1 and G 2 thenwegetthefollowing subcases.Since G 1 and G 2 haveallpossible2-leavessothelastandrst K 3 softhe underlying2-pathofboth G 1 and G 2 musthavea2-leafinitsinterioredge.Hencethe lastaddedvertexontherightendof G 1 if G 1 isassumedtobeconstructedfromthe leftandtherstaddedvertexontheleftendof G 2 if G 2 isassumedtobeconstructed fromtheleftmustallbenon-probes.Theonlypossiblestructuresthatcanformhere areillustratedintheFigure4.102.Inothercasesweeithergetadjacenciesbetween non-probeverticeswhichgivesustheforbiddensubgraph E 2 ;E 3 ;E 4 ;E 5 ;E 10orget an F 2intheunderlying2-path.Theunitproberepresentationsofthestructuresin 163

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Figure4.102aregiveninthesamegure.Thesestructuresarepossibletorepresent becausetherightend2-leaffrom G 1 isnon-adjacenttotheleftend2-leafof G 2 as showninthesamegure.Furthermoreitcanbeeasilyseenfromthesestructuresthat wecannothaveany2-leafontheinterioredgesoftheintermittent2-pathinpicture 1ofthesameguresincethatwillformeitheran E 3oran E 4andwealsocanhave exactlyone2-leafononeinterioredgeoftheintermittent2-pathasshowninpicture 2otherwisewewillgettheforbiddensubgraph E 2 ;E 5 ;E 6 ;E 10.Pictures1and2are representedbydoingtherepresentationof G 1 rstasintheFigure4.97followedby thestraight2-pathasintheFigure4.103andthennallydoingtherepresentationof G 2 asintheFigures4.89,4.90,4.91.Picture4istheonlydierentcaseherewhose representationisgiveninthesamegure.Thisstructurecontainsastraight2-path oflength10and3fromtheend4-fansand4fromtheintermittentstraight2-path oflength4.Heretoowedotherepresentationof G 1 rstasinpicture1ofthe Figure4.97followedbythestraight2-pathasintheFigure4.103andthennally doingtherepresentationof G 2 asintheFigures4.89,4.90,4.91sothattheleftend 2-leavesof G 2 havedistinctendpointsinalloftheserepresentationswecanmake theend2-leavesoftheunderlying2-pathhavedistinctendpoints.Furthermoreon anypartofthestructurewhichhasatleastsixconsecutive K 3 s,ifedge e k hasa 2-leafthenitsnearest2-leafmustbeontheedge e k +3 , e k )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 otherwisewewillget theforbiddensubgraphs.Soavoidingsuch2-leaveswecaneasilygettheunitprobe intervalrepresentationofthislastsubcase. Case6 n> 4:Ifthereexistsmorethanfour K 3 'sbetween G 1 and G 2 thenasbeforewegetthreesubcases.Thisisthecasewheretheintermittentstraight2-path isoflengthgreaterthan4.Itcanbeconcludedfromthepreviouscasestudythat ifweconstructallthepossiblestructuresaswedidforcase5wecaneithergetany structurewhoserepresentationcanbedonebyrepresenting G 1 rstasinFigure4.97 followedbythestraight2-pathasinFigure4.103andthennallydoingtherepresen164

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Figure4.108: n =0: I 3 tationof G 2 asinFigures4.89,4.90,4.91orstructuresjoinedbystraight2-pathsof lengthsatleast11atleast5fromtheintermittentstraight2-pathsand3eachfrom the4-fansontwoends.The2-leavesontheinterioredgesofthestraight2-pathsare presentsothatnoforbiddensubgraphsisformed.Soasstatedbefore,ifedge e k has a2-leafthenitsnearest2-leafmustbeontheedge e k +3 , e k )]TJ/F17 7.9701 Tf 6.586 0 Td [(3 .Theunitprobeinterval representationofsuchstructuresarepossiblebecausetheend2-leavesofthe4-fans ontwoendswhicharepartsofthestraight2-pathcanbemadetohavedistinctend pointsbythesecondpicturesofFigures4.89,4.90andeasytoseein4.91andby representationinFigure4.97andtheintermittentstraight2-pathcanberepresented asinFigure4.103. Bythesameargumentusedintheproofsofndinganunitprobeintervalrepresentationof I 1 and I 2 ,itcanbeconcludedthattheresultsalsoholdsformergeof I 3 . Note:Thisnoteisvalidforalltheabove3theorems.Letusconsidersome B ij where B i and B j arejoinedbysomestraight2-pathoflength p .Supposethatthis B ij failstohavearepresentationatthemergeduetoadjacencyofnon-probes.Itis 165

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Figure4.109: n =1: I 3 Figure4.110: n =2: I 3 166

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Figure4.111: Interior2-caterpillarswithfewer2-leaves easytoseethatthissamegraphwillbeunitprobeintervalrepresentableif,atthe merge,one2-leafistakenofromeither B i or B j sometimesfromboth B i and B j as inpicture4oftheFigure4.111,andthelasttwo4-fansofthat B i or B j orboth B i and B j areisomorphictoeitherastaircaseoravertex-consecutivestructuresothat thenon-proberestrictionfromitsend2-leafisremovedandtheirintervalscanbe madetohavedistinctendpoints.Intheothercaseswegettheforbiddensubgraphs asseeninFigure4.87. Thesestructurescanbelookedasmergeofinterior2-caterpillarswithfewer2leavesor I i , i =1 ; 2 ; 3withfewer2-leaves.Theirrepresentationsaregiveninthe Figure4.111.Itisworthmentioningherethatwecanalsogetotherkindsofinterior 2-caterpillarslikeinterior2-caterpillarswhicharejustsubgraphsof I 1 ;I 2 ;I 3 .Allthese structureswillalsohaveunitprobeintervalrepresentationsincefromalltheprevious representationsweknowthatallthe2-leavesarenon-probesandtheirremovalfrom therepresentationwillnotaecttheintervallayoutoftheunderlying2-path. 167

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Wenowhaveacharacterizationforinterior2-caterpillarsisomorphicto I 1 ;I 2 ;I 3 whichareunitprobeintervalgraphs.Sinceaninterior2-caterpillariseitherisomorphictooneoftheabove I i where i =1 ; 2 ; 3,theirsubgraphsor I i 'swithfewer 2-leaves,wecannoweasilycharacterizeinterior2-caterpillarswhichareunitprobeintervalgraphs.Thefollowingtheoremgivesacharacterizationofinterior2-caterpillars whichareunitprobeintervalgraphs. Theorem4.7.7 Aninterior 2 -caterpillarisaunitprobeintervalgraphifandonly ifitdoesnotcontainE1,E2,E3,E4,E5,E6,E7,E8,E9,E9,E10,Group-E11,F2, F3,F4,F5,F6,F7,F8,F9,F10,F11asgeneratedsubgraphs. Proof: Let G beaninterior2-caterpillarwhichisaunitprobeintervalgraph. Since E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11donothaveunitprobeintervalrepresentation, G cannothave themasinducedsubgraphs. Fortheotherdirection,let G beanyinterior2-caterpillarwithout E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asgeneratedsubgraphs.Anyinterior2-caterpillarcanbeconsideredasgroupsof edgeorvertexconsecutive4-fansjoinedtogetherbystraight2-pathswithallpossible 2-leavesorfewer2-leaves.So G canbeassumedtobegroupsofedgeorvertex consecutive4-fansjoinedtogetherbystraight2-pathswithallpossibleorfewer2leavessuchthatnowhere E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11areformedasinducedsubgraphs.Thus G iseither I 1 ;I 2 ;I 3 ; theirsubgraphsor I i 'si=1,2,3withfewer2-leaveswithout E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11.Thereforebythepreviousthreetheoremsandtheabovenoteitmust haveaunitprobeintervalrepresentation. 168

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Thenexttheoremderivesarelationshipbetween2-treeunitprobeintervalgraphs andinterior2-caterpillars. Theorem4.7.8 A2-treeisaunitprobeintervalgraphifandonlyifitisaninterior 2-caterpillarwithoutE1,E2,E3,E4,E5,E6,E7,E8,E9,E9,E10,Group-E11,F1, F2,F3,F4,F5,F6,F7,F8,F9,F10,F11asgeneratedsubgraphs. Proof: Thisproofisanextensionofpreviousresults.Werstprovethenecessarypartrst.Let G be2-treeUPIG.ByTheorem4.3.3wehavethat G mustbean interior2-caterpillar.Since E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11arenotUPIG, G cannotcontain anyofthemasinducedsubgraphs.So G mustbeaninterior2-caterpillarwithout E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11. Fortheotherdirectionletusnowassumethat G isa2-treeinterior2-caterpillar without E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asinducedsubgraphs.Bytheprevioustheorem,an interior2-caterpillarisaunitprobeintervalgraphifandonlyifitdoesnotcontain E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asgeneratedsubgraphs.Since G isaninterior2-caterpillarwithout E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11itmustbeaunitprobeintervalgraph. Finallythenexttheoremgivesacompletecharacterizationof2-treeswhichare unitprobeintervalgraphs.Thetotalnumberofforbiddensubgraphsis27. Theorem4.7.9 A2-treeisaunitprobeintervalgraphifandonlyifitdoesnot containE1,E2,E3,E4,E5,E6,E7,E8,E9,E9,E10,Group-E11,F1,F2,F3,F4, F5,F6,F7,F8,F9,F10,F11asgeneratedsubgraphs. 169

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Proof: Let G bea2-tree.letusrstassumethatitisaunitprobeinterval graph.Since E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11donothaveunitprobeintervalinterval representation, G cannotcontainthemasinducedsubgraphs. Fortheotherdirectionletusnowassumethat G isa2-treewithout E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asinducedsubgraphsandthat G isnotaunitprobeintervalgraph.Then bytheprevioustheorem G isnotaninterior2-caterpillarwithout E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11asgeneratedsubgraphs.Butitisgiventhat G doesnotcontainanyinduced E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10,GroupE 11, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11.So G mustnotbeaninterior2-caterpillar.Soiteitheris anon-interior2-caterpillarornota2-caterpillaratall.Thus G eitherhasan F 1 or G has B 1 ;B 1 0 ;B 2 ;B 3asinducedsubgraphs.But G cannothavean F 1asan inducedsubgraphsoitmusthave B 1 ;B 1 0 ;B 2 ;B 3asinducedsubgraphs.Itiseasy toseefromFigure4.9that B 1 ;B 1 0 have E 1and B 2 ;B 3have F 1initrespectivelyas inducedsubgraphs.Thisimpliesthat G has F 1or E 1asinducedsubgraphswhichis acontradiction.Hence G mustbeaunitprobeintervalgraph. Conclusion:Theprecedingresultscomplimenttheexistingcharacterizationsofsubclassesof probeintervalgraphswhichincludecycle-free[41],unitcycle-free[15]andbipartite unit[10].TheseresultsalsoaddtothespiritoftheresultsofCorneilandPrzulj[34] inshowingthatthecompletecharacterizationofevenrestrictedsubclassesofprobe intervalgraphswillbeachallengingproblem. Someprogress,however,hasbeenmadeinfndingclassesofprobeintervalgraphs whichwilladmittoanalbeitcomplicatecharacterizationviaforbiddeninduced subgraphs.SeeforexampletherecentworkofBrown,Busch,andIsaak[5],andof 170

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Brown,Flesch,andLundgren[7]. 171

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