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PDF of the Phd Thesis Durham E-Theses Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireexive Graphs THOMAS, DANIEL,JAMES,RHYS How to cite: THOMAS, DANIEL,JAMES,RHYS (2020) Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireexive Graphs , Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/13671/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs Daniel Thomas A Thesis presented for the degree of Doctor of Philosophy Department of Computer Science Durham University United Kingdom June 2020 Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs Daniel Thomas Submitted for the degree of Doctor of Philosophy June 2020 Abstract: In this thesis we investigate three different aspects of graph theory. Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdős, Ko and Rado to graphs. Our main results are a proof of an Erdős-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber. Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically per- fect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses. Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture. Declaration The work in this thesis is based on research carried out in the Department of Computer Science at Durham University. No part of this thesis has been submitted elsewhere for any degree or qualification, and the work is all my own work unless specified to the contrary in the text. Copyright © 2020 Daniel Thomas . “The copyright of this thesis rests with the author. No quotation from it should be published without the author’s prior written consent and information derived from it should be acknowledged.” Acknowledgements To start with I want to thank the Durham School of Engineering and Computing Sciences for having me, and allowing me the use of their facilites which are first rate. I am also grateful to the department for allowing me to demonstrate in the MCS problem classes, which has been a valuable learning experience. I owe a debt of gratitude to my supervisor Matthew. For accepting me as his student, and for his patient support and encouragement throughout my studies, even in times where progress was difficult. He has been consistently available for meetings with myself, which have been informative, enjoyable and with an open-minded atmosphere. I am also thankful for his comments on the draft of this thesis which have been invaluable. Thanks are also due to Carl Feghali, with whom I enjoyed a fruitful collaboration. Finally I am extremely grateful to my family, especially my mother and father, who have supported me throughout. Is it one living being, Which has separated in itself? Or are these two, who chose To be recognized as one? — from Gingo Biloba by Goethe Dedicated to Lynn Thomas and Stephen Thomas Contents Abstract ii List of Figures xi 1 Introduction 1 1.1 Basic Definitions . .3 1.2 Overview of Thesis . .7 2 Independent Sets In Trees and the EKR Theorem 15 2.1 Introduction . 15 2.2 The Main Results . 18 2.3 Depth-two Claws . 19 2.4 Elongated Claws with Short Limbs . 24 2.5 Star Systems and Counterexamples to Conjecture 2.1.7 . 27 2.6 More general counterexamples to Conjecture 2.1.7 . 32 2.7 Trees with Multiple Large Non-Leaf Centred Stars . 35 2.8 Conjecture 2.1.7 is true for Caterpillars . 40 2.9 Remark on the number of independent r-sets in paths . 45 2.10 Remark on Disjoint Unions Of Complete Graphs . 47 Contents ix 3 Edge Modification in Perfect Graphs 48 3.1 Introduction . 48 3.1.1 Subclasses of the perfect graphs and algorithmic properties . 53 3.1.2 Main results . 56 3.2 Some Definitions and a proof of the Perfect Graph Theorem . 60 3.3 Edge Modification and the Parameters α and ω ........ 65 3.3.1 Connectivity . 69 3.3.2 Pancritical Graphs . 72 3.4 Edge Modification and The SPGT . 74 3.4.1 Dichotomy in Odd Holes and Odd Antiholes . 76 3.4.2 Minimum Imperfect Berge Graphs and Dichotomy . 77 3.4.3 Existence Of Berge-Critical Edges . 80 3.4.4 Berge-Critical Edges Contained in a P4 and Berge-Critical Edges Incident with a Vertex . 81 3.4.5 Edges that are Critical and Berge-Critical . 85 3.4.6 Triangles and Berge-Critical edges . 86 3.4.7 Distance 2 and Berge-Critical Edges . 86 3.4.8 Distance 2, Holes, and Star-Cutsets . 87 3.4.9 A Note on Distance 2 and a Property of Odd Holes . 89 3.4.10 Concluding Remarks to Section 3.4 . 90 3.5 Freedom Games . 91 3.5.1 Freedom and Perfect Graphs . 92 3.5.2 An Application to Critically and Cocritically Perfect Graphs 105 3.5.3 A Bipartite Reconfiguration-type Graph . 107 3.5.4 Concluding Remarks to Section 3.5 . 109 Contents x 4 Bireflexive Graphs, Homomorphisms, and Modular Decomposition111 4.1 Introduction . 112 4.2 Modular Decomposition: Modules, Skeletons and Primality . 114 4.3 Graph Homomorphisms . 120 4.3.1 Homomorphic Equivalence and Cores . 124 4.3.2 The Direct Product . 125 4.4 Bireflexive Graphs . 126 4.4.1 Bireflexive Graphs and Lexicographic Products . 128 4.4.2 Analogues of Theorems 4.3.3, 4.3.4 and 4.3.6 . 130 4.5 Analogues Of Colouring and Perfect Graphs . 131 4.6 The Dual Of The Reconstruction Conjecture . 134 Bibliography 146 List of Figures 2.1 T (5, 2, 3) ...................... 28 3.1 The first three examples of minimally imperfect graphs are given by C5, C7 and C7.................... 52 3.2 The unfolded tetrahedron. 68 3.3 The cycle on 5 vertices . 73 3.4 The Paley graph P (13) ................. 74 3.5 No way to add an edge to 2K2 without getting a P4 ...... 93 4.1 A biconnected graph with a module formed by the vertices labelled {1, 2, 3}....................... 116 Chapter 1 Introduction A graph is a collection of vertices, which may be adjoined pairwise by edges. In every- day language, a graph is a network. Graph theory is a branch of mathematics which studies graphs, which model the real-life networks one sees in various everyday and practical settings: friendship networks, telephone networks, and electrical networks would be some prominent examples. One simple and pervasive example of a graph is the kind that graph theorists call trees, so named because they model the branching and vascular structure of trees (and other plants). This type of graph occurs widely: in biological settings such as family trees and phylogeny, in computing settings in searching and storing data, or in communications settings in Morse code (as binary trees). We shall see some problems on trees in Chapter 2. As with any mathematical subject, graph theory has its ultimate origins in some kind of problem that needed solving. The historical origin of graph theory is thus traced to a seemingly frivolous problem that was considered, and solved, by Euler in 1736 [27]. The locals of Königsberg wondered how to traverse each of the seven bridges of Königsberg exactly once: Euler gave a simple argument that it was not possible as too many of the land masses of Königsberg could be reached from each other in an odd number of ways. Chapter 1. Introduction 2 At the time, Euler considered his theorem on the bridges of Konigsberg to be one about the “geometry of position”. The phrase “graph” was originally coined by James Joseph Sylvester in 1878 in the paper “Chemistry and Algebra” on molecular diagrams [91]. Indeed there has been a great deal of interaction between the study of graphs and chemistry, as the bonding structure of molecules can be modelled by graphs. Another problem which is both deeply entwined with the history and developement of graph theory, and also central in modern applications of graph theory, is that of graph colouring.
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