Abstract
Kayles is a combinatorial game on graphs. Two players select alternatingly a vertex from a given graph G—a chosen vertex may not be adjacent or equal to an already chosen vertex. The last player that can select a vertex wins the game. The problem to determine which player has a winning strategy is known to be PSPACE-complete. Because of certain characteristics of the Kayles game, it can be analyzed with Sprague–Grundy theory. In this way, we can show that the problem is polynomial time solvable on graphs with a bounded asteroidal number. It is shown that the problem can be solved in O(n3) time on cocomparability graphs and circular arc graphs, and in O(n1+1/log3)=O(n1.631) time on cographs.
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