Games on interval and permutation graph representations

We describe combinatorial games on graphs in which two players antagonistically build a representation of a subgraph of a given graph. We show that for a large class of these games, determining whether a given instance is a winning position for the next player is PSPACE-hard. In contrast, we give polynomial time algorithms for solving some versions of the games on trees

HABILITATION A DIRIGER DES RECHERCHES Graphes et jeux combinatoires

On considère généralement que la théorie des graphes est née au 18e siècle, et qu'elle connaît un essor significatif depuis les années 1960. L'avènement de la théorie des jeux combinatoires est quant à lui plus récent (fin des années 1970). Ce domaine reste alors moins exploré dans la littérature, et de nombreuses études sur des techniques générales de résolution sont toujours actuellement en cours de construction. Dans ce mémoire, je propose plusieurs tours d'horizons à propos de problématiques bien ciblées de ces deux domaines.Dans un premier temps, je m'interroge sur la complexité des règles de jeux de suppression de tas. Il s'avère que dans la littérature, la complexité d'un jeu est souvent définie comme la complexité algorithmique d'une stratégie gagnante. Cependant, il peut aussi avoir du sens de s'interroger sur la nature des règles de jeu. Un premier pas dans cette direction a été fait avec l'introduction du concept de jeu dit invariant. On notera au passage que certains résultats obtenus ont mis en exergue des liens entre combinatoire des mots et stratégie gagnante d'un jeu. Dans un deuxième chapitre, j'aborde les jeux sous l'angle des graphes. Deux aspects sont considérés:* Un graphe peut être vu comme un support de jeu. Le cas du jeu de Nim et ses variantes sur les graphes y est examiné.* Certaines problématiques standard de théorie des graphes peuvent être transformées dans une version ludique. C'est d'ailleurs un objet d'étude de plus en plus prisé par la communauté. Nous détaillerons le cas des jeux de coloration sommet.Enfin, le dernier chapitre se concentre sur deux nouvelles variantes de problématiques issues de la théorie des graphes: le placement de graphes et les colorations distinguantes. J'en profite pour faire un état de l'art des principaux résultats sur ces deux domaines

Exact algorithms for Kayles

In the game of Kayles, two players select alternatingly a vertex from a given graph G, but may never choose a vertex that is adjacent or equal to an already chosen vertex. The last player that can select a vertex wins the game. In this paper, we give an exact algorithm to determine which player has a winning strategy in this game. To analyze the running time of the algorithm, we introduce the notion of a K-set: a nonempty set of vertices W ⊆ V is a K-set in a graph G = ( V , E ) , if G [ W ] is connected and there exists an independent set X such that W = V − N [ X ]. The running time of the algorithm is bounded by a polynomial factor times the number of K-sets in G. We prove that the number of K-sets in a graph with n vertices is bounded by O(1.6052^n). A computer-generated case analysis improves this bound to O(1.6031^n) K-sets, and thus we have an upper bound of O(1.6031^n) on the running time of the algorithm for Kayles. We also show that the number of K-sets in a tree is bounded by n ⋅ 3 n / 3 and thus Kayles can be solved on trees in O(1.4423^n) time. We show that apart from a polynomial factor, the number of K-sets in a tree is sharp. As corollaries, we obtain that determining which player has a winning strategy in the games G_avoid ( POS DNF 2 ) and G_seek ( POSDNF_3 ) can also be determined in O(1.6031^n) time. In G_avoid(POSDNF_2) , we have a positive formula F on n Boolean variables in Disjunctive Normal Form with two variables per clause. Initially, all variables are false, and players alternately set a variable from false to true; the first player that makes F true loses the game. The game G_seek ( POSDNF 3 ) is similar, but now there are three variables per clause, and the first player that makes F true wins the game

Exact algorithms for Kayles

In the game of Kayles, two players select alternatingly a vertex from a given graph G, but may never choose a vertex that is adjacent or equal to an already chosen vertex. The last player that can select a vertex wins the game. In this paper, we give an exact algorithm to determine which player has a winning strategy in this game. To analyze the running time of the algorithm, we introduce the notion of a K-set: a nonempty set of vertices W ⊆ V is a K-set in a graph G = ( V , E ) , if G [ W ] is connected and there exists an independent set X such that W = V − N [ X ]. The running time of the algorithm is bounded by a polynomial factor times the number of K-sets in G. We prove that the number of K-sets in a graph with n vertices is bounded by O(1.6052^n). A computer-generated case analysis improves this bound to O(1.6031^n) K-sets, and thus we have an upper bound of O(1.6031^n) on the running time of the algorithm for Kayles. We also show that the number of K-sets in a tree is bounded by n ⋅ 3 n / 3 and thus Kayles can be solved on trees in O(1.4423^n) time. We show that apart from a polynomial factor, the number of K-sets in a tree is sharp. As corollaries, we obtain that determining which player has a winning strategy in the games G_avoid ( POS DNF 2 ) and G_seek ( POSDNF_3 ) can also be determined in O(1.6031^n) time. In G_avoid(POSDNF_2) , we have a positive formula F on n Boolean variables in Disjunctive Normal Form with two variables per clause. Initially, all variables are false, and players alternately set a variable from false to true; the first player that makes F true loses the game. The game G_seek ( POSDNF 3 ) is similar, but now there are three variables per clause, and the first player that makes F true wins the game