COMPLEXITY OF GENERALIZED COLOURINGS OF CHORDAL GRAPHS by Juraj Stacho M.Sc., Comenius University, Slovakia, 2005 a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the School of Computing Science c Juraj Stacho 2008 SIMON FRASER UNIVERSITY Spring 2008 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Juraj Stacho Degree: Doctor of Philosophy Title of thesis: Complexity of Generalized Colourings of Chordal Graphs Examining Committee: Dr. Joseph G. Peters Chair Dr. Pavol Hell, Senior Supervisor Dr. Ramesh Krishnamurti, Supervisor Dr. G´abor Tardos, SFU Examiner Dr. Ross M. McConnell, External Examiner, Computer Science Department, Colorado State University Date Approved: ii Abstract The generalized graph colouring problem (GCOL) for a fixed integer k, and fixed classes of graphs ,..., (usually describing some common graph properties), is to decide, for P1 Pk a given graph G, whether the vertex set of G can be partitioned into sets V1,...,Vk such that, for each i, the induced subgraph of G on V belongs to . It can be seen that GCOL i Pi generalizes many natural colouring and partitioning problems on graphs. In this thesis, we focus on generalized colouring problems in chordal graphs. The struc- ture of chordal graphs is known to allow solving many difficult combinatorial problems, such as the graph colouring, maximum clique and others, in polynomial, and in many cases in linear time. Our study of generalized colouring problems focuses on those problems in which the sets are characterized by a single forbidden induced subgraph. We show, that Pi for k = 2, all such problems where the forbidden graphs have at most three vertices are polynomial time solvable in chordal graphs, whereas, it is known that almost all of them are NP -complete in general. On the other hand, we show infinite families of such problems which are NP -complete in chordal graphs. By combining a polynomial algorithm and an NP -completeness proof, we answer a question of Broersma, Fomin, Neˇsetˇril and Woeginger about the complexity of the so-called subcolouring problem on chordal graphs. Additionally, we explain, how some of these results generalize to particular subclasses of chordal graphs, and we show a complete forbidden subgraph characterization for the so-called monopolar partitions of chordal graphs. Finally, in the last part of the thesis, we focus on a different type of colouring problem – injective colouring. We describe several algorithmic and (in-)approximability results for injective colourings in the class of chordal graphs and its subclasses. In the process, we correct a result of Agnarsson et al. on inapproximability of the chromatic number of the square of a split graph. Keywords: chordal graphs; graph colouring; subcolouring; injective colouring; forbidden subgraph characterization; polynomial time algorithms Subject Terms: Graph Theory; Graph Coloring; Graph Algorithms; Perfect Graphs; Homomorphisms (Mathematics); Graph Grammars iii Mojim rodiˇcom Milanovi a Darine To my parents Milan and Darina iv Acknowledgments The author would like to seize the opportunity to express his deepest gratitude to his thesis advisor Dr. Pavol Hell for his guidance, enthusiasm, his profound insight, and broad knowledge, and most importantly for his perpetual support during author’s graduate studies at Simon Fraser University. The author also wishes to thank his parents for their unconditional support throughout his entire academic studies and wishes to dedicate this thesis to them. v Contents Approval ii Abstract iii Dedication iv Acknowledgments v Contents vi List of Figures viii List of Tables ix List of Algorithms ix 1 Introduction 1 1.1 Definitions...................................... 4 1.2 Chordalgraphs ................................... 7 1.3 Relatedclasses .................................. 11 1.4 Polynomialtimecases ............................. 15 2 Tools 22 2.1 MatrixPartitions................................ 22 2.2 Dynamicprogramming.............................. 24 2.3 Greedyalgorithms ................................ 32 2.4 GraphGrammars.................................. 34 2.5 Treewidth and Monadic Second Order Logic . ...... 38 3 Polar colourings of Chordal Graphs 40 3.1 MonopolarChordalGraphs . 40 3.2 PolarChordalGraphs .............................. 53 vi 4 Forbidden Subgraphs for Monopolarity 63 4.1 Thetwo-connectedcase .. .. .. .. .. .. .. .. 64 4.2 Thegeneralcase .................................. 68 5 Subcolourings of Chordal Graphs 89 5.1 2-subcolourings ................................. 90 5.1.1 Subcolouringdigraph . 90 5.1.2 Algorithm.................................. 96 5.2 3-subcolourings ................................. 101 5.3 k-subcolourings...................................105 6 Pk-transversals of Chordal Graphs 108 6.1 Stable P4-transversals ...............................108 6.2 Forcinggraphs ................................... 112 6.3 Stable P5-transversals ...............................117 6.4 Pj-free Pk-transversals ...............................120 6.5 Kj-free Pk-transversals...............................124 7 Other cases 126 7.1 StronglyChordalGraphs . 135 7.2 ChordalComparabilityGraphs . 138 8 Injective Colourings 143 8.1 Basicproperties................................. 144 8.2 Hardnessandapproximationresults . 146 8.3 Exactalgorithmicresults . 151 8.3.1 Injectivestructure . .. .. .. .. .. .. .. 153 8.3.2 Computing χi(G)inchordalgraphs . .154 8.3.3 Bridgelesschordalgraphs . 156 8.3.4 Perfectly tree-dominated graphs . 157 9 Conclusions 160 Bibliography 162 vii List of Figures 1.1 All minimal precoloured forbidden induced subgraphs of split graphs. 18 2.1 The stubborn problem, its complement, and two NP -complete cases. 23 2.2 An example of a hyperedge replacement grammar generating all partial 3-trees. 36 3.1 Rules1-8. ..................................... 44 3.2 An illustration of the monopolarity recognition algorithm. ........... 45 4.1 All minimal list non-extendable forbidden induced subgraphs for monopolar- ityof2-connectedchordalgraphs.. 65 4.2 TherulesofthegrammarΓ.. 69 4.3 Additional rules of the grammar Γ′. ....................... 69 4.4 Example of a derivation of the grammar. ...... 70 4.5 Hypergraphs (part 1) for the proof of Theorem 4.9. ........ 83 4.6 Hypergraphs (part 2) for the proof of Theorem 4.9. ........ 84 4.7 Hypergraphs (part 3) for the proof of Theorem 4.9. ........ 86 5.1 Illustrating the case when there is an arc u, v in (G). ........... 91 h i Dc 5.2 The graph H and sample 3-subcolourings of H. ................103 5.3 The graph (X)withasample3-subcolouring. .103 G 5.4 The graphs Fk for small k, and the construction from Theorem 5.15. 106 6.1 The graphs Gi and Yj................................109 6.2 The cotree for the graph G S . ........................111 i − i 6.3 Forcing graphs for the Pk-transversalproblem. .112 k 6.4 Forcing graphs Hj fortheproofofTheorem6.1. .115 k k 6.5 Forcing graphs Bj and Dj for the proof of Theorem 6.1. 116 ′ ′ 6.6 The graphs Yj and Gi. ..............................117 ′ ′′ ′′′ 6.7 The graphs G (G ) and G for the Pj-free Pk-transversal problem. 120 ∗ 6.8 The graph G for the Kj-free Pk-transversal problem. 124 7.1 Matrices for selected GCOL problems in chordal graphs. ..........126 7.2 Transforming a transitive orientation into a uniform transitive orientation. 140 viii List of Tables 1.1 A summary of complexities of selected graph problems. .......... 14 9.1 A summary of complexity results in chordal graphs and general graphs. 160 List of Algorithms 1.1 Lexicographic Breadth-First search. ......... 10 3.1 Monopolar graph recognition. ...... 43 3.2 Constructingreducedlists. ...... 43 3.3 TheRules1-8andtheBlockRule. 44 3.4 Extracting a monopolar partition. ....... 46 3.5 Testingwhetheracliqueisnice. ...... 61 3.6 Polargraphrecognition.. ..... 62 5.1 The test for 2-subcolourability of G......................... 99 5.2 The test whether Pu,v is (z, w)-colourable.. .100 5.3 The test whether T is ( )colourable... .. .. .. .. .. ..100 x − 5.4 The test whether Tx is(+)colourable.. .101 7.1 Cardinality Lexicographic Breadth-First search. .............139 ix Chapter 1 Introduction The graph colouring problem is probably one of the oldest problems of graph theory. Here, the task is to label the vertices of a given graph using as few colours as possible in such a way that no adjacent vertices receive identical labels. The origins of this problem can be traced all the way back to the famous problem of four colours, formulated by Francis Guthrie in 1852, which asks whether any map of countries can be coloured using at most four colours in such a way that no two countries sharing a boundary use the same colour. As it turned out later, this is just a special case of the graph colouring problem in planar graphs. Since then, graph colouring and its variants have appeared in many different guises in combinatorics, optimization, discrete geometry, set theory, and many other branches of discrete mathematics. Also, many practical problems such as time tabling, sequencing, and scheduling are related to the problem of graph colouring. Unfortunately, graph colouring is one of the first known NP -complete problems. This tells us that it is unlikely that an efficient algorithm for graph colouring exists, where, of course, by efficient, we mean running in polynomial time (possibly also randomized). This, however, does not mean that the problem