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Combinatorial Games on Graphs Gabriel Renault Combinatorial games on graphs Gabriel Renault To cite this version: Gabriel Renault. Combinatorial games on graphs. Other [cs.OH]. Université Sciences et Technologies - Bordeaux I, 2013. English. NNT : 2013BOR14937. tel-00919998 HAL Id: tel-00919998 https://tel.archives-ouvertes.fr/tel-00919998 Submitted on 17 Dec 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. N◦ d’ordre: 4937 THÈSE présentée à L’UNIVERSITÉ BORDEAUX 1 École Doctorale de Mathématiques et Informatique de Bordeaux par Gabriel Renault pour obtenir le grade de DOCTEUR SPÉCIALITÉ : INFORMATIQUE Jeux combinatoires dans les graphes Soutenue le 29 novembre 2013 au Laboratoire Bordelais de Recherche en Informatique (LaBRI) Après avis des rapporteurs : Michael Albert Professeur Sylvain Gravier Directeur de recherche Brett Stevens Professeur Devant la commission d’examen composée de : Examinateurs Tristan Cazenave Professeur Éric Duchêne Maître de conférence Rapporteurs Sylvain Gravier Directeur de recherche Brett Stevens Professeur Directeurs de thèse Paul Dorbec Maître de conférence Éric Sopena Professeur ii Remerciements iii Remerciements Je voudrais d’abord remercier les rapporteurs et membres de mon jury de s’être intéressés à mes travaux, d’avoir lu ma thèse, assisté à ma soutenance et posé des questions évoquant de futurs axes de recherche sur le sujet. Merci beaucoup à Michael Albert, Sylvain Gravier et Brett Stevens d’avoir pris le temps de lire ma thèse en détail et exprimé des remarques pertinentes et constructives sur leurs rapports. Merci beaucoup à Tristan Cazenave d’avoir accepté de présider mon jury. Merci beaucoup à Éric Duchêne d’être venu de Lyon pour ma soutenance. Je tiens à remercier plus particulièrement mes directeurs de thèse Paul Dorbec et Éric Sopena. Travailler avec eux a toujours été un plaisir, ils ont un don pour déterminer quels problèmes m’intéresseraient particulièrement. Ils m’ont fait confiance en acceptant de m’encadrer pour mon stage de M2, puis pour ma thèse. Je les remercie aussi pour toutes les opportunités qu’ils m’ont apportées, que ce soit des expériences d’enseignement un peu différentes au sein de Math en Jeans et Maths à Modeler, des séjours de recherche ou des conférences. L’échange avec eux a aussi été très agréable sur le plan humain, c’était sympathique de pouvoir aussi parler en dehors d’un contexte de travail. Je voudrais également remercier les personnes avec lesquelles j’ai eu l’occasion de travailler, en particulier Boštjan, Éric, Gašper, Julien, Re- becca, Richard, Sandi et Simon. C’est toujours intéressant de se pencher sur un problème avec quelqu’un qui peut avoir une vision un peu différente et d’arriver à une solution grâce à ces deux points de vue. J’ai aussi eu la chance de participer aux séminaires du groupe de tra- vail Graphes et Applications. Je remercie ses membres et plus particulière- ment ceux qui ont été doctorants en même temps que moi, avec lesquels j’ai eu le plus d’échange, Adrien, Clément, Émilie Florent, Hervé, Noël Petru, Pierre, Quentin, Romaric, Sagnik, Thomas et Tom. C’était aussi un plaisir de cotoyer les membres permanents, André, Arnaud, Cyril, David, Frantisek, Frédéric, Mickael, Nicolas, Nicolas, Olivier, Olivier, Philippe et Ralf. Je remercie aussi les chercheurs en graphes et jeux que j’ai eu la chance de rencontrer en conférence ou séjour de recherche, parmi lesquels Aline, An- dré, Ararat, Benjamin, Guillem, Guillaume, Jean-Florent, Julien, Laetitia, Laurent, Louis, Marthe, Neil, Nicolas, Paul, Reza, Théophile et Urban. J’ai eu la chance d’enseigner à l’Université Bordeaux 1, et je voudrais re- mercier les enseignants qui m’ont accompagné pour ce parcours, notamment Aurélien, Christian, David, Frédéric, Frédérique, Jean-Paul, Pascal, Pierre, Pierre et Vincent. Je tiens également à remercier Marie-Line avec laquelle j’ai pu encadrer l’activité Math en Jeans à l’université. Je remercie aussi toute l’équipe administrative du LaBRI pour leur effi- cacité, leur amabilité et leur bonne humeur, en particulier ceux avec lesquels iv Remerciements j’ai eu le plus d’occasions de discuter, Brigitte, Cathy, Isabelle, Lebna, Maïté et Philippe. Je voudrais remercier les doctorants qui ont participé à faire vivre le bureau 123 via des débats et discussions, en particulier Amal, Benjamin, Christian, Christopher, Edon, Eve, Florian, Jaime, Jérôme, Lorijn, Lucas, Matthieu, Nesrine, Omar, Oualid, Pierre, Sinda, Tom, Tung, Van-Cuong, Wafa et Yi. La vie au labo aurait été différente sans les activités organisées par l’AFoDIB, et les sorties entre doctorants. Je remercie leurs participants les plus fréquents, parmi ceux qui n’ont pas encore été cités, Abbas, Adrien, Allyx, Anaïs, Anna, Carlos, Cedric, Cyril, David, Dominik, Etienne, Fouzi, Jérôme, Lorenzo, Martin, Matthieu, Noémie-Fleur, Razanne, Rémi, Simon, Srivathsan, Thomas, Vincent et Xavier. J’ai aussi eu des contacts en dehors de la sphère de l’université. Je voudrais remercier mes amis avec lesquels mes sujets de discussions variaient un peu en comparaison, ou qui m’ont permis de vulgariser mes travaux, en particulier Alexandre, Amélie, Benjamin, Cécile, Clément, Emeric, Ena, Guilhem, Irène, Jonas, Kévin, Laure, Nolwenn, Ophélia, Pierre, Rémi et Ro- main. Je citerai aussi les emails du RatonLaveur qui pouvaient participer à me mettre de bonne humeur. Je voudrais aussi remercier ma famille pour des raisons similaires, ainsi que pour m’avoir fait confiance et encouragé dans la voie que je voulais suivre. Je remercie mes parents, mes frères et sœur, mes grands-parents, oncles, tantes, cousins et cousines. Un remerciement spécial à Xiaohan qui a donné une autre dimension à la vulgarisation en me poussant à la réaliser en anglais. Résumé v Jeux combinatoires dans les graphes Résumé : Dans cette thèse, nous étudions les jeux combinatoires sous différentes contraintes. Un jeu combinatoire est un jeu à deux joueurs, sans hasard, avec information complète et fini acyclique. D’abord, nous regardons les jeux impartiaux en version normale, en particulier les jeux VertexNim et Timber. Puis nous considérons les jeux partisans en version normale, où nous prouvons des résultats sur les jeux Timbush, Toppling Dominoes et Col. Ensuite, nous examinons ces jeux en version misère, et étudions les jeux misères modulo l’univers des jeux dicots et modulo l’univers des jeux dead-endings. Enfin, nous parlons du jeu de domination qui, s’il n’est pas combinatoire, peut être étudié en utilisant des outils de théorie des jeux combinatoires. Mots-clés : jeux combinatoires, graphes, jeux impartiaux, jeux partisans, version normale, version misère, jeu de domi- nation vi Abstract Combinatorial games on graphs Abstract: In this thesis, we study combinatorial games under different conventions. A combinatorial game is a finite acyclic two-player game with complete information and no chance. First, we look at impartial games in normal play and in particular at the games VertexNim and Timber. Then, we consider partizan games in normal play, with results on the games Timbush, Toppling Dominoes and Col. Next, we look at all these games in misère play, and study misère games modulo the dicot universe and modulo the dead-ending universe. Finally, we talk about the domination game which, despite not being a combinatorial game, may be studied with combinatorial games theory tools. Keywords: combinatorial games, graphs, impartial games, partizan games, normal convention, misère convention, dom- ination game Contents vii Contents 1 Introduction 1 1.1 Definitions............................. 2 1.1.1 Combinatorial Games . 2 1.1.2 Graphs .......................... 6 2 Impartial games 13 2.1 VertexNim ............................ 14 2.1.1 Directed graphs . 16 2.1.2 Undirected graphs . 21 2.2 Timber .............................. 26 2.2.1 General results . 27 2.2.2 Trees............................ 32 2.2.2.1 Paths . 39 2.3 Perspectives............................ 40 3 Partizan games 43 3.1 Timbush ............................. 44 3.1.1 General results . 45 3.1.2 Paths ........................... 49 3.1.3 Black and white trees . 54 3.2 Toppling Dominoes ...................... 61 3.2.1 Preliminary results . 63 3.2.2 Proof of Theorem 3.27 . 65 3.3 Col ................................ 72 3.3.1 General results . 76 3.3.2 Known results . 80 3.3.3 Caterpillars . 93 3.3.4 Cographs ......................... 96 3.4 Perspectives............................ 98 4 Misère games 101 4.1 Specificgames........................... 104 4.1.1 Geography .......................105 4.1.2 VertexNim .......................111 4.1.3 Timber ..........................113 4.1.4 Timbush .........................116 4.1.5 Toppling Dominoes ..................117 viii Contents 4.1.6 Col ............................119 4.2 Canonical form of dicot games . 126 4.2.1 Definitions and universal properties . 127 4.2.2 Canonical form of dicot games . 130 4.2.3 Dicot misère games born by day 3 ...........136 4.2.3.1 Dicot games born by day 3 in the general universe..................... 143 4.2.4 Sums of dicots can have any outcome . 146 4.3 A peek at the dead-ending universe . 147 4.3.1 Preliminary results . 149 4.3.2 Integers and other dead ends . 151 4.3.3 Numbers.......................... 153 4.3.3.1 The misère monoid of Q2 ...........153 4.3.3.2 The partial order of numbers modulo E . 158 4.3.4 Zeros in the dead-ending universe . 160 4.4 Perspectives............................ 161 5 Domination Game 167 5.1 About no-minus graphs . 169 5.2 The domination game played on unions of graphs . 172 5.2.1 Union of no-minus graphs .
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