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Characterising Parameterized Graph Classes: Certifying Algorithms For Characterising parameterized graph classes: Certifying algorithms for fixed-parameter tractable problems Samuel Stuart Wilson Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds School of Computing May 2015 The candidate confirms that the work submitted is his/her own, except where work which has formed part of a jointly authored publication has been included. The contribution of the candidate and the other authors to this work has been explicitly indicated below. The candidate confirms that appropriate credit has been given within the thesis where reference has been made to the work of others. Some parts of the work presented in Chapters 5, 6 and 7 have been published in the following articles: M¨ullerH; Wilson S An FPT certifying algorithm for the vertex-deletion problem in: Lecroq T; Mouchard L (editors) Combinatorial Algorithms, 24th International Workshop, IWOCA 2013, pp.468-472. Springer. 2013. M¨ullerH; Wilson S Characterising subclasses of perfect graphs with respect to partial or- ders related to edge contraction in: Cornelissen K; Hoeksma R; Hurink J; Manthey B (editors) Proceedings of the 12th Cologne-Twente Workshop on Graphs and Combinato- rial Optimization (CTW 2013), vol. WP 13-01, pp.183-186. Centre for Telematics and Information Technology. 2013. The above publications are primarily the work of the second author. The first author is the primary supervisor of the candidate and therefore discussions on techniques, correctness and presentation were had. The contribution of the first author is that of a primary supervisor. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis maybe published without proper acknowledgement. ©2015 The University of Leeds and Samuel Stuart Wilson Acknowledgements I would like to thank many people for supporting me throughout the duration of my PhD. I hope that people know that their support was vital for the completion of my thesis and I request that the omission of acknowledgement of any person who feels they have supported my studies does not diminish my gratitude for their support. Firstly, I would like to thank my supervisory team; Haiko M¨ullerand Kristina Vuˇskovi´cfor their support and encouragement throughout my time at the University of Leeds. In particular I would like to thank Haiko for the countless hours he has spent expanding my enjoyment of mathematics and the patient support he has provided ensuring that I meet his expectations. I would like to thank Kristina for ensuring I understand the role of rigour and clarity in my research and also for her encouragement for me to teach during my studies. I am exceedingly honoured and proud to say that I have been supervised by them. I would like to thank the members of the School of Computing for creating a happy and productive working environment. I have had many stimulating discussions on various topics which have given me insight into new areas of study. I would like to extend my gratitude to the Algorithms and Complexity research group for revealing a multitude of interesting theoretical problems to me. I would like to acknowledge the EPSRC and the School of Computing for funding my research through a doctoral training grant, without which pursing a PhD would not have been possible. Lastly I would like to thank my friends and family who have supported me throughout my PhD; my family for their moral support and my friends for listening to me speak, often at length, about my work. I would like to recognise my brother, Ed, for his continued support throughout by academic studies. I would like to express my complete appreciation of the mem- bers of my research lab for their general support and their willingness to engage in random, but fruitful, discussion about anything. In particularly, Elaine and Sarah who have been invaluable, managing to keep my work/life balance, mostly, to a reasonable level. Without Elaine my thesis would certainly not have reached this point. Finally I would like to thank the examiners for their effort. Abstract In this thesis we consider the problem of characterising parameterized graph classes. The parameterized graph classes we consider are the graph classes where there exists a small number of modifications that yields a well studied graph class. This type of graph class appears in many fields such as in computational biology, data sciences and communication networks. We prove, under some weak assumptions, that many of the classes can be characterised by a finite minimal forbidden set. We also provide a formalisation of properties of partial orders and demonstrate that many of the results in the literature, such as the well-quasi ordering theorems of Ding, of Damaschke, and of Robertson and Seymour can also be applied to other partial orders. We prove that it is possible to form a lattice structure from the set of all partial orders on finite graphs and introduce a set of tools for inferring properties of those partial orders. The results presented in this thesis have a number of consequences. As a result of the finite characterisation of the parameterized graph classes, we develop a generic algorithm for enumerating the minimal forbidden set for each class where the results may be applied. The enu- meration enables the development of structural theorems concerning the parameterized graph classes which leads to the development of efficient algorithms. The results presented in this thesis also have applications in the field of certifying algorithms. We provide motivation for the development of fixed-parameter certifying algorithms and provide the first fixed-parameter certifying algorithm. We apply the results to the vertex deletion problem, showing a general construction for a fixed-parameter certifying algorithm to recognise the parameterized graph classes we consider. The results of this thesis also provide characterisations for a set of subclasses of perfect graphs with respect to partial orders that include edge contraction. Contents 1 Introduction 1 2 Notation 5 2.1 General . .5 2.1.1 Graph operations . .6 2.2 Relations . .8 2.2.1 Well-quasi ordering . .9 2.2.2 Partial orders on graphs . 10 2.3 Graph classes . 12 2.3.1 Parameterized graph classes . 20 2.4 Width parameters . 21 2.4.1 Connections between width parameters . 23 2.5 Fixed-parameter tractability . 23 2.6 Certifying algorithms . 24 3 Related Work 27 3.1 Overview . 27 3.2 Partial orders . 28 3.2.1 Kruskal's tree theorem . 32 3.2.2 Graph minor theorem . 33 3.2.3 Well-quasi ordering . 40 3.3 Graph classes . 42 3.4 Graph modification problems . 46 3.5 Fixed-parameter algorithms . 51 3.5.1 Bounded search tree . 52 3.5.2 Kernelization . 53 3.5.3 Iterative compression . 54 3.5.4 Meta theorems . 54 3.6 Certifying algorithms . 56 i ii CONTENTS 4 Properties of Partial Orders 63 4.1 Lattice of partial orders . 63 4.2 Partial orders & parameterized graph classes . 76 4.3 Summary . 83 5 Characterising almost graphs 85 5.1 Constructing a bound for almost graphs . 86 5.1.1 Overview . 92 5.1.2 C+kv ...................................... 93 5.1.3 Worked example for the induced subgraph relation . 96 5.1.4 C+ke ...................................... 98 5.2 Summary . 99 6 Applications 101 6.1 Enumerating the minimal forbidden sets . 101 6.2 Certifying fixed-parameter algorithms . 110 6.3 Summary . 119 7 Edge contraction 121 7.1 Contraction minors . 121 7.2 False twin minors . 132 7.3 Containment Complexity . 137 7.3.1 Contractibility . 137 7.3.2 False twin minors . 142 7.4 Edge contraction and well-quasi ordering . 144 7.5 Summary . 145 8 Topological Minors 147 8.1 Coincidence of C+kv .................................. 149 8.2 C is well-quasi ordered by 6t ............................. 151 8.3 C has a single minimal forbidden graph . 151 8.4 Summary . 154 9 Conclusions 157 9.1 Summary . 157 9.2 Future work . 158 Appendices 161 A Small graphs 163 B Psuedocode 165 List of Figures 2.1 Complete graphs: K3;K4 and Kn ........................... 13 2.2 Path graphs: P2;P3 and Pn .............................. 14 2.3 Cycle graphs: C3;C4 and Cn ............................. 14 2.4 Star graphs: K1;3;K1;4 and K1;k ........................... 14 2.5 2 × 2-grid, 2 × 3-grid and 3 × 3-grid respectively. 15 2.6 Wheel graphs: W3;W4 and Wk ............................ 15 2.7 Minimal forbidden planar graphs: K5;K3;3 ..................... 15 2.8 An example of a perfect elimination ordering . 16 2.9 Minimal forbidden cograph: P4 ............................ 16 2.10 Example of an intersection model . 17 n+1 n 2.11 Minimal forbidden induced subgraphs for interval graphs. Cn,T2, XF2 , XF3 for n ≥ 0. ........................................ 17 2.12 Minimal forbidden interval graphs with respect to induced minors . 17 2.13 Example of permutation model . 18 2.14 Minimal forbidden induced subgraphs for trivially perfect graphs; P4;C4..... 18 2.15 Minimal forbidden induced subgraphs for co-trivially perfect graphs; 2K2;C4... 19 2.16 Minimal forbidden induced subgraphs for threshold graphs; 2K2;C4;P4..... 19 2.17 Minimal forbidden induced subgraphs for split graphs; 2K2;C4;C5......... 20 3.2 Minor minimal obstructions for treewidth 3. 35 3.3 Outline of the graph minor theorem . 39 3.4 Antichains in the set of all graphs with respect to 6i ................ 40 3.5 Expected behaviour of a conventional program . 57 3.6 Behaviour of a certifying program . 58 4.2 Hasse diagram of a lattice of partial orders . 72 4.6 An example of C+1v not being closed with respect to 6c.............. 79 4.7 An example of Forb(C+1v) not being finite where Forb(C) is finite with respect to 6it..........................................
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