THÈSE pour obtenir le grade de DOCTEUR DE LA COMMUNAUTÉ UNIVERSITÉ GRENOBLE ALPES Spécialité : Informatique et Mathématiques Appliquées Arrêté ministériel : 25 mai 2016 Présentée par Andrea Munaro Thèse dirigée par András Sebo˝ et Matejˇ Stehlík préparée au sein du Laboratoire G-SCOP dans l’école doctorale Mathématiques, Sciences et Technologies de l’Information, Informatique (MSTII) Sur quelques invariants classiques et nouveaux des hypergraphes Thèse soutenue publiquement le 1er décembre 2016, devant le jury composé de: Dieter Rautenbach Professeur des Universités, Universität Ulm, Rapporteur Nicolas Trotignon Directeur de Recherche CNRS, ENS de Lyon, Rapporteur András Gyárfás Senior Researcher, Alfréd Rényi Institute of Mathematics, Examinateur Frédéric Maffray Directeur de Recherche CNRS, Univ. Grenoble Alpes, Laboratoire G-SCOP, Examinateur et Président du jury András Sebo˝ Directeur de Recherche CNRS, Univ. Grenoble Alpes, Laboratoire G-SCOP, Directeur de thèse Matejˇ Stehlík Maître de Conférences, Univ. Grenoble Alpes, CNRS, Laboratoire G-SCOP, Co-encadrant de thèse Acknowledgements First of all, I am grateful to my supervisors András Sebo˝ and Matˇej Stehlík. Thank you for your support, encouragement, patience and for all the helpful suggestions. I also appreciated a lot the freedom you gave me during these intense three years. Then, I would like to thank the other members of my thesis committee: András Gyárfás, Frédéric Maffray, Dieter Rautenbach and Nicolas Trotignon. Thank you for accepting to be part of it and for travelling to Grenoble the day of my defense. I would also like to thank all the members of the “Optimisation Combinatoire” group, my fellow Ph.D. students, in particular Lucas, Quentin and Yohann, and my officemates Kaveh, Saleh and Van-Thao. A special thanks goes to Javi, Miguel and Teri for the nice reunions we had. I am also grateful to Fadila Messaoud-Djebara, Marie-Josephe Perruet, Christine Rouzier and Myriam Torlini for their support in administrative matters. Finally, I want to express my gratitude to my family, my friends and to you, Marie. i Contents Introduction 1 1 Preliminaries7 2 Generalized Line Graphs 19 2.1 Introduction..................................... 19 2.2 Line graphs of subcubic triangle-free graphs................... 22 2.2.1 Characterizations.............................. 22 2.2.2 Independence number........................... 25 2.2.3 Complexity of Feedback Vertex Set.................... 31 2.2.4 Complexity of Hamiltonian Cycle and Hamiltonian Path........ 37 2.3 k-Line graphs.................................... 40 3 Approximate Min-Max Theorems 45 3.1 Introduction..................................... 45 3.2 Bounded clique cover................................ 49 3.2.1 θ-Bounding functions............................ 51 3.2.2 Algorithmic aspects of Clique Cover.................... 60 3.3 On Tuza’s Conjecture................................ 62 3.3.1 Graphs with edges in few triangles.................... 68 3.3.2 θ-Bounding functions for classes related to triangle graphs....... 70 3.4 On Jones’ Conjecture................................ 73 3.4.1 Subcubic graphs.............................. 77 4 VC-Dimension 83 4.1 Introduction..................................... 83 4.2 Bounds on the VC-dimension........................... 87 4.3 The decision problem................................ 93 4.3.1 Graphs of bounded clique-width..................... 95 4.3.2 Subclasses of planar graphs........................ 97 4.4 Complexity dichotomies in monogenic classes.................. 100 5 Boundary Classes For NP-Hard Graph Problems 105 5.1 Introduction..................................... 105 5.2 Hamiltonian Cycle Through Specified Edge and Hamiltonian Path....... 110 5.3 Feedback Vertex Set................................ 117 5.4 Connected Dominating Set............................. 122 5.5 Connected Vertex Cover.............................. 125 Bibliography 129 Index Of Notions Not Defined In Chapter1 141 iii List of Figures 1.1 Some special graphs.................................9 2.1 The minimal forbidden induced subgraphs for the class of line graphs..... 20 2.2 A subcubic triangle-free graph G with 10 edges and α0(G) = 3.......... 20 2.3 The possible subgraphs induced by the closed neighbourhood of a vertex of a (K4; claw; diamond)-free graph........................... 25 2.4 Illustration of three cases in the proof of Theorem 2.2.6............. 27 2.5 The hexagon implant operation........................... 33 2.6 The gadget in the proof of Theorem 2.2.15.................... 33 2.7 The gadget in the proof of Theorem 2.2.17.................... 35 2.8 A graph showing that the approximation factor of Algorithm1 gets arbitrarily close to 2....................................... 36 2.9 The construction of the graph G0 in the proof of Theorem 2.2.23........ 38 2.10 Twin-C5........................................ 41 2.11 The two not necessarily induced subgraphs of G corresponding to an induced P4 in T (G)...................................... 42 3.1 The graphs G11 and G13............................... 49 3.2 The difficult blocks in Theorem 3.2.6........................ 51 3.3 The graph G8..................................... 58 3.4 The dodecahedron graph.............................. 80 4.1 Upper and lower leaves in the proof of Theorem 4.2.1.............. 88 4.2 The neighbourhood of w0 in the proof of Theorem 4.2.1............. 90 4.3 The different constructions of a tree with at least A + k 1 leaves in the proof j j − of Theorem 4.2.1................................... 91 4.4 The construction in the proof of Theorem 4.3.1.................. 95 4.5 The construction in the proof of Theorem 4.3.6.................. 99 5.1 A tribranch Yi;j;k................................... 110 5.2 A caterpillar with hairs of arbitrary length..................... 110 5.3 The construction in the proof of Lemma 5.2.6................... 112 5.4 The operation T k................................... 116 5.5 The graph Hi..................................... 117 5.6 The graph Si;j;k;`................................... 118 5.7 The vertex stretching operation........................... 120 5.8 The construction in the proof of Theorem 5.4.1.................. 123 5.9 The different spanning trees in the proof of Theorem 5.4.1........... 124 5.10 The operation Ap................................... 126 0 5.11 The graph Hi..................................... 127 v Introduction In general, the structure of a (hyper)graph can be complicated, both from a combinatorial and algorithmic point of view. On the other hand, it is often the case that restricting the range makes possible to obtain certain properties. In this thesis, we consider several (hy- per)graph parameters and study whether restrictions to subclasses of hypergraphs allow to obtain desirable combinatorial or algorithmic properties. Most of the parameters we consider are special instances of packings and transversals of hypergraphs and we examine essentially three kinds of “desirable” properties: Existence of a large packing; • Existence of an upper bound for the transversal number in terms of the packing number; • Polynomial-time decidability of the packing and transversal numbers and of some (hy- • per)graph properties. Along the way, we also consider a prominent measure of the “complexity” of a hypergraph: its VC-dimension. Let us now properly define the notions introduced above. A packing of a hypergraph = (V; E) is a set of pairwise disjoint edges of . A related notion is that of a transversal H H (also known as hitting set or covering) of , which is a subset X V intersecting each edge H ⊆ of . Clearly, every hypergraph has a packing (the empty set) and so it is natural to look H for maximum packings, namely packings with as many edges as possible. Similarly, every hypergraph has a transversal (the vertex set) and we are interested in minimum transversals, namely transversals with as few vertices as possible. The packing number ν( ) is the number H of edges in a packing of of maximum size (a maximum packing) and the transversal number H τ( ) is the number of vertices in a transversal of of minimum size (a minimum transversal). H H In Chapter2, we study the packing number of two hypergraphs arising from graphs. The first is obtained by considering the graph itself, and so a packing is what is usually known as a matching. The other hypergraph is the dual of the clique hypergraph of a graph G, where the clique hypergraph is the hypergraph having as vertices the vertices of G and as edges the maximal cliques of G. It is easy to see that a packing of this hypergraph is nothing but an independent set of G. A natural relation between the notions of matching and independent set arises when considering the class of line graphs1: Indeed, there is a bijection between the matchings of a graph and the independent sets of its line graph. Line graphs constitute a rich and ubiquitous class of graphs, introduced by Whitney [183]. In Chapter2, we concentrate on a subclass of line graphs: line graphs of subcubic triangle-free graphs. In particular, we provide several characterizations of this class and we study the independence number of its graphs. The famous Brooks’ Theorem asserts that every connected graph G which is neither a complete graph nor an odd cycle must be ∆(G)-colourable and so α(G) V (G) =∆(G). ≥ j j Following this result, several authors considered the problem of finding tight lower bounds for the independence number of graphs having bounded maximum degree