Kayles is the game, where two players alternately choose a vertex that has not been chosen before nor is adjacent to an already chosen vertex from a given graph. The last player that choses a vertex wins the game. We show, with help of Sprague-Grundy theory, that the problem to determine which player has a winning strategy for a given graph, can be solved in O(n³) time on interval graphs, on circular arc graphs, on permutation graphs, and on co-comparability graphs and in O(n^1.631) time on cographs. For general graphs, the problem is known to be PSPACE-complete, but can be solved in time polynomial in the number of isolatable sets of vertices of the graph
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