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Walks, Transitions and Geometric Distances in Graphs

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Walks, Transitions and Geometric Distances in Graphs THESE` pr´esent´ee `a L’UNIVERSITE´ DE BORDEAUX ECOLE´ DOCTORALE DE MATHEMATIQUES´ ET D’INFORMATIQUE par Thomas Bellitto POUR OBTENIR LE GRADE DE DOCTEUR SPECIALIT´ E´ : INFORMATIQUE Walks, Transitions and Geometric Distances in Graphs. Soutenue publiquement le 27 aoˆut2018 apr`es avis des rapporteurs : Jørgen Bang-Jensen, Professeur, University of Southern Denmark Sylvain Gravier, Directeur de recherche, Universit´eJoseph Fourier devant la commission d’examen compos´ee de : Christine Bachoc, professeure, Universit´ede Bordeaux, directrice de th`ese Jørgen Bang-Jensen, professeur, University of Southern Denmark, rapporteur Sylvain Gravier, directeur de recherche, Universit´eJoseph Fourier, rapporteur Mickael Montassier, professeur, Universit´ede Montpellier, examinateur Arnaud Pecherˆ , professeur, Universit´ede Bordeaux, directeur de th`ese Eric´ Sopena, professeur, Universit´ede Bordeaux, pr´esident du jury 2018 Walks, Transitions and Geometric Distances in Graphs. Abstract This thesis studies combinatorial, algorithmic and complexity aspects of graph theory problems, and especially of problems related to the notions of walks, transitions and distances in graphs. We first study the problem of traffic monitoring, in which we have to place as few censors as possible on the arcs of a graph to be able to retrace walks of objects. The characterization of instances of practical interests brings us to the notion of forbidden transitions, which strengthens the model of graphs. Our work on forbidden-transition graphs also includes the study of connecting transition sets, which can be seen as a translation to forbidden-transition graphs of the notion of spanning trees. A large part of this thesis focuses on geometric graphs, which are graphs whose vertices are points of the real space and whose edges are determined by geometric distance between the vertices. This graphs are at the core of the famous Hadwiger- Nelson problem and are of great help in our study of the density of sets avoiding distance 1 in various normed spaces. We develop new tools to study these problems and use them to prove the Bachoc-Robins conjecture on several parallelohedra. We also investigate the case of the Euclidean plane and improve the bounds on the density of sets avoiding distance 1 and on its fractional chromatic number. Finally, we study the complexity of graph homomorphism problems and estab- lish dichotomy theorems for the complexity of locally-injective homomorphisms to reflexive tournaments. Keywords Graphs, walks, forbidden transitions, graph homomorphisms, indepen- dence number, geometric distances, sets avoiding distance 1, NP-completeness Affiliation University of Bordeaux, CNRS, LaBRI, UMR 5800, F-33400 Talence, France Walks, Transitions and Geometric Distances in Graphs. iii iv Thomas Bellitto Marches, Transitions et Distances G´eom´etriques dans les Graphes. R´esum´e Cette th`ese ´etudie les aspects combinatoires, algorithmiques et la complexit´ede probl`emes de th´eorie des graphes, et tout sp´ecialement de probl`emes li´es aux notions de marches, de transitions et de distance dans les graphes. Nous nous int´eressons d’abord au probl`eme de traffic monitoring, qui consiste `a placer aussi peu de capteurs que possible sur les arcs d’un graphe de fa¸con `apouvoir reconstituer des marches d’objets. La caract´erisation d’instances int´eressantes dans la pratique nous am`ene `ala notion de transitions interdites, qui renforce le mod`ele de graphe. Notre travail sur les graphes `atransitions interdites comprend aussi l’´etude de la notion d’ensemble de transitions connectant, que l’on peut voir comme l’analogue en terme de transitions de la notion d’arbre couvrant. Une partie importante de cette th`ese porte sur les graphes g´eom´etriques, qui sont des graphes dont les sommets sont des points de l’espace r´eel et dont les arˆetes sont d´etermin´ees par les distances g´eom´etriques entre les sommets. Ces graphes sont au cœur du c´el`ebre probl`eme de Hadwiger-Nelson et nous sont d’une grande aide dans notre ´etude de la densit´edes ensembles qui ´evitent la distance 1 dans plusieurs types d’espaces norm´es. Nous d´eveloppons des outils pour ´etudier ces probl`emes et les utilisons pour prouver la conjecture de Bachoc-Robins sur plusieurs parall´elo`edres. Nous nous penchons aussi sur le cas du plan euclidien et am´eliorons les bornes sur la densit´edes ensembles ´evitant la distance 1 et sur son nombre chromatique fractionnaire. Enfin, nous ´etudions la complexit´ede probl`emes d’homomorphismes de graphes et ´etablissons des th´eor`emes de dichotomie sur la complexit´edes homomorphismes localement injectifs vers les tournois r´eflexifs. Mots-cl´es Graphes, marches, transitions interdites, homomorphismes de graphes, nombre de stabilit´e, distances g´eometriques, ensemble ´evitant la distance 1, NP- compl´etude Affiliation Universit´ede Bordeaux, CNRS, LaBRI, UMR 5800, F-33400 Talence, France Walks, Transitions and Geometric Distances in Graphs. v vi Thomas Bellitto Acknowledgements Je commence bien sˆur cette section par un tr`es grand merci `a Arnaud, qui m’encadre depuis mon stage de master et qui m’a ´enorm´ement aid´e`atrouver ce que je voulais faire et `aarriver l`ao`uj’en suis. Merci pour ton soutien, ton aide et tes conseils dans tellement de domaines : la recherche, l’enseignement, la r´edaction, la recherche de stages puis de postdocs, les candidatures... Merci aussi pour ta confi- ance et la libert´eque tu m’as laiss´ee pendant ma th`ese pour trouver les probl`emes sur lesquels je voulais travailler et les directions que je voulais explorer. Merci surtout pour tout le temps que tu as toujours r´eussi `atrouver dans ton emploi du temps charg´epour ´ecouter, relire et essayer de comprendre mes id´ees tordues. Un grand merci aussi `aChristine, qui m’a beaucoup aid´e`aenrichir mon domaine de recherche, qui m’a fait d´ecouvrir des probl`emes passionnants et qui m’a ´enorm´ement aid´e`ales relier `ace que je savais faire et `apartir dans la bonne direction. I also thank Sylvain Gravier and Jørgen Bang-Jensen for accepting to read the manuscript and Eric´ Sopena and Mickael Montassier for being part of the jury. I would also like to thank Jørgen Bang-Jensen again and Anders Yeo for choosing me for an open postdoc position in their team. I am really excited for this next important step of my journey. Of course, I would like to thank my co-authors and all the people I have worked with and without whose ideas this thesis would probably never have existed. Arnaud et Christine, une fois encore. Philippe, mon fr`ere de th`ese, avec qui ¸ca a toujours ´et´eun grand plaisir de travailler ou de discuter. Gary, of course; I had an amazing time in Victoria thanks to you and you have done a lot to help me since. Thank you to my other co-authors from Victoria too: Chris, Stefan and Feiran. Merci `a Benjamin avec qui j’ai beaucoup aim´etravailler et ´echanger. Merci aussi `aAntoine avec qui ¸ca a ´et´eun plaisir de travailler ces derniers mois. Thank you to my previous advisors: Gunnar Klau, Tobias Marschall, Alexander Sch¨onhuth, David Coudert, Nicolas Nisse and Endre Boros. Merci ´evidemment `atoute l’´equipe de th´eorie des graphes du LaBRI pour de nombreux ´echanges passionnants et pour un cadre de travail et une ambiance tr`es agr´eables. Merci aussi `amon ´equipe INRIA, R´ealopt, et `a mes co-bureaux, Henri et Th´eo. Merci `aHerv´eet `atoute l’´equipe de Maths `aModeler pour cette exp´erience tr`es enrichissante. Un grand merci aussi `atous mes coll`egues d’´equipes p´edagogiques. Je pense vii notamment `aCarole, Olivier, Feri et Cyril avec qui ¸ca a ´et´eun grand plaisir de travailler. Merci aussi `atous ceux que j’ai oubli´es et qui m’ont aid´e`a arriver l`ao`uje suis aujourd’hui. Merci `atous mes amis, du labo et d’ailleurs. Un ´enorme merci tout partic- uli`erement `aTh´eo et Antonin pour leur soutien et pour tous les bons moments pass´es ensemble ces derni`eres ann´ees. Merci enfin `ama famille et surtout `ama m`ere et mon fr`ere pour leur soutien inconditionnel depuis toutes ces ann´ees. viii Thomas Bellitto Contents Introduction 9 1 Preliminaries 17 1.1 Fundamentals of graph theory ...................... 18 1.1.1 Core definitions .......................... 18 1.1.2 Homomorphisms and colouring ................. 23 1.1.3 Special vertex sets ........................ 26 1.1.4 Hypergraphs ........................... 28 1.2 Elements of complexity ......................... 30 1.2.1 P, NP and polynomial reductions ................ 30 1.2.2 3-SAT and NP-completeness .................. 31 1.2.3 Approximations .......................... 33 1.3 Walks, connectivity and transitions ................... 33 1.3.1 Walks and connectivity in usual graphs ............ 33 1.3.2 Forbidden-transition graphs ................... 36 1.4 Polytopes and lattices .......................... 38 1.4.1 Norms and distances ....................... 38 1.4.2 Lattices .............................. 40 1.4.3 Polytopes ............................. 41 1.4.4 Classification of the parallelohedra in dimension 2 and 3 . 43 1.5 Rational languages and automata .................... 45 1.5.1 Rational languages ........................ 46 1.5.2 Automata and recognition .................... 47 1.6 Linear programming ........................... 50 1.6.1 Definitions ............................ 50 1.6.2 Integer linear programming ................... 52 2 Separating codes and traffic monitoring 55 2.1 Introduction ................................ 55 2.2 The traffic monitoring problem ..................... 56 2.2.1 Definition ............................. 56 2.2.2 Limitations of the existing separation models ......... 57 2.3 A new model of separation: separation on a language ......... 59 2.3.1 Presentation of the problem ................... 59 2.3.2 Expressiveness of the model ................... 59 ix CONTENTS 2.4 Separation of a finite set of walks .................... 60 2.5 Separation of walks with given endpoints ..............
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