Journal of Algorithms
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Theoretical Computer Science
Exact Exponential Algorithms
Kayles on the way to the stars
CG'04 Proceedings of the 4th international conference on Computers and Games
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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 analyse 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], where N[X] is the union of X and the set of all vertices adjacent to at least one vertex of X. The running time of the algorithm is bounded by a polynomial factor times the number of K-sets in G. We show that the number of K-sets in a graph with n vertices is bounded by O(1.6052n), and thus we have an algorithm for Kayles with running time O(1.6052n). We also show that the number of K-sets in a tree is bounded by n ·3n/3 and thus Kayles can be solved on trees in O(1.4423n) time. We show that apart from a polynomial factor, the number of K-sets in a tree is sharp.