On Treewidth and Graph Minors Daniel John Harvey Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy February 2014 Department of Mathematics and Statistics The University of Melbourne Produced on archival quality paper ii Abstract Both treewidth and the Hadwiger number are key graph parameters in structural and al- gorithmic graph theory, especially in the theory of graph minors. For example, treewidth demarcates the two major cases of the Robertson and Seymour proof of Wagner's Con- jecture. Also, the Hadwiger number is the key measure of the structural complexity of a graph. In this thesis, we shall investigate these parameters on some interesting classes of graphs. The treewidth of a graph defines, in some sense, how \tree-like" the graph is. Treewidth is a key parameter in the algorithmic field of fixed-parameter tractability. In particular, on classes of bounded treewidth, certain NP-Hard problems can be solved in polynomial time. In structural graph theory, treewidth is of key interest due to its part in the stronger form of Robertson and Seymour's Graph Minor Structure Theorem. A key fact is that the treewidth of a graph is tied to the size of its largest grid minor. In fact, treewidth is tied to a large number of other graph structural parameters, which this thesis thoroughly investigates. In doing so, some of the tying functions between these results are improved. This thesis also determines exactly the treewidth of the line graph of a complete graph. This is a critical example in a recent paper of Marx, and improves on a recent result by Grohe and Marx. By extending the techniques used, we also determine the treewidth of the line graph of a complete multipartite graph, up to lower order terms in general, and exactly whenever the complete multipartite graph is regular. This generalises a result by Lucena. We also determine a lower bound on the treewidth of any line graph; this result is similar to a question about the Hadwiger number of line graphs posed by Seymour, which was recently proven by DeVos et al.. Finally, we prove a result on the treewidth of the Kneser graph; in doing so we also prove a generalisation of the famous Erd}os-Ko-Rado Theorem. The Hadwiger number of a graph is the size of its largest complete minor. One of the most important conjectures in modern mathematics is Hadwiger's Conjecture, which conjectures that the Hadwiger number of a graph is at least its chromatic number. A related question is determining what lower bound on the average degree is required to iii iv ABSTRACT ensure the existence of a Kt-minor (or, more generally, an H-minor for any graph H). The Kt-minor case has been thoroughly studied, and independently answered by Kostochka and Thomason. In this thesis we answer a slightly different question and present an algorithm for finding, in O(n) time, an H-minor forced by high average degree. Finally, this thesis determines a weakening of Hadwiger's Conjecture on the class of circular arc graphs, an interesting generalisation of the class of interval graphs, and in the process of doing so, proves some useful results about linkages in interval graphs. Declaration This is to certify that: (i) the thesis comprises only my original work towards the PhD except where indicated in the Preface, (ii) due acknowledgement has been made in the text to all other material used, and (iii) the thesis is less that 100,000 words in length, exclusive of tables, maps, bibliogra- phies and appendices. v vi DECLARATION Preface No work presented in this thesis has been submitted for any other kind of qualification, and no work in this thesis was carried out prior to PhD enrolment. Other than in Chapter1 and Chapter2, all results presented are new. (Some of the results presented in Chapter2, an investigation into parameters tied to treewidth, are improvements on previously presented results. These results are noted in the text.) Chapters2,3,4,5,6 and8 were carried out in collaboration with my supervisor, David Wood. Chapter7 is the result of joint work with Vida Dujmovi´c,Gwena¨el Joret, Bruce Reed and David Wood. An important example in Chapter5 was provided by Bruce Reed. The results of Chapter7 extend results which have been recently published in the SIAM Journal on Discrete Mathematics [26]. The results in Chapter6 have recently appeared in the Electronic Journal of Combinatorics [44]. The results in Chapter3 have been accepted by the Journal of Graph Theory [45]. (An earlier version of this paper, including the results of Chapter4, is available on the arXiv [42].) Chapter2 is also available on the arXiv [43]. In all cases I, Daniel Harvey, was the primary and corresponding author. vii viii PREFACE Acknowledgements I'd like to acknowledge the following individuals for their assistance during my PhD can- didature. Firstly, I'd like to thank my supervisor David Wood for the enormous amount of assistance provided throughout my time as a PhD student. For their guidance, I'd like to thank the members of my Advisory Committee|Sanming Zhou, Peter Forrester and Graham Farr. I'd like to thank my collaborators Vida Dujmovi´c,Gwena¨elJoret and Bruce Reed for their work on the paper which would become Chapter7. Further thanks to my officemates and fellow students Michael Payne, Guangjun Xu and Ricky Rotheram for their advice and support. I also wish to acknowledge the several reading and seminar groups that ran during my candidature. Firstly, the Graph Theory Reading Group, and its current leader Arun Mani. Secondly, the Discrete Structures and Algorithms seminar group. Finally, the Theoretical Research in Computer Science (TRICS) group, run by Tony Wirth and (previously) Kerri Morgan. Thanks to Alex Scott for pointing out references [36, 37, 99] in Chapter6. Thanks also to Jacob Fox for helpful conversations with regards to Chapter2. A final note of thanks to my parents David and Jan Harvey for their emotional support during the last four years. ix x ACKNOWLEDGEMENTS Table of Contents Abstract iii Declarationv Preface vii Acknowledgements ix Table of Contents xi List of Figures xv 1 Introduction and Literature Review1 1.1 Graph Minors...................................1 1.2 Treewidth.....................................3 1.3 Hadwiger's Conjecture.............................. 10 1.4 A Unifying Example............................... 17 I Treewidth 19 2 Parameters Tied to Treewidth 21 2.1 Introduction.................................... 21 2.2 Basics....................................... 23 2.3 Brambles..................................... 24 2.4 k-Trees and Chordal Graphs........................... 25 2.5 Separators..................................... 26 2.6 Branchwidth and Tangles............................ 30 2.7 Tree Products................................... 34 2.8 Linkedness..................................... 36 2.9 Well-linked and k-Connected Sets........................ 38 xi xii TABLE OF CONTENTS 2.10 Grid Minors.................................... 40 2.11 Grid-like Minors................................. 41 2.12 Fractional Open Problems............................ 44 3 Treewidth of the Line Graph of a Complete Graph 45 3.1 Introduction.................................... 45 3.2 Line-Brambles and the Treewidth Duality Theorem.............. 46 3.3 Proof of Result.................................. 48 4 Treewidth of the Line Graph of a Complete Multipartite Graph 51 4.1 Introduction.................................... 51 4.2 Line-Brambles of a Complete Multipartite Graph............... 52 4.3 Path Decompositions............................... 63 5 Treewidth of General Line Graphs 73 5.1 Introduction.................................... 73 5.2 The General Lower Bound............................ 74 5.3 The General Upper Bound and Extensions................... 79 6 Treewidth of the Kneser Graph and the Erd}os-Ko-RadoTheorem 81 6.1 Introduction.................................... 81 6.2 Basic Definitions and Preliminaries....................... 82 6.3 Upper Bound for Treewidth........................... 84 6.4 Separators in the Kneser Graph......................... 86 6.5 Lower Bound for Treewidth when k = 2.................... 91 6.6 A Weaker Lower Bound for Treewidth..................... 92 6.7 Open Questions.................................. 95 II Graph Minors 99 7 Finding a Minor Quickly in Graphs with High Average Degree 101 7.1 Introduction.................................... 101 7.2 Algorithm..................................... 101 7.3 Correctness of Algorithm............................ 103 7.4 Time Complexity................................. 104 8 Hadwiger's Conjecture for Circular Arc Graphs 105 8.1 Introduction.................................... 105 TABLE OF CONTENTS xiii 8.2 Preliminaries................................... 105 8.3 Special Path Sets................................. 109 8.4 Colouring G .................................... 116 8.5 Extensions..................................... 121 9 Linkages in Interval Graphs 123 9.1 Introduction.................................... 123 9.2 \Selection Sort" Paths in the Power of a Path................. 125 9.3 Improved Linkages in Interval Graphs..................... 128 9.4 Hadwiger Number of the Power of a Cycle................... 135 Bibliography 137 Index 146 xiv TABLE OF CONTENTS List of Figures 1.1 An example graph and tree decomposition...................4 1.2 Catlin's counterexample to Haj´os'Conjecture.................. 14 2 1.3 C9 represented as a circular arc graph...................... 17 2.1 The graph 4;2..................................