Games without repetitions on digraphs (Q1325234)

We consider two-person games of type Nim on a finite directed graph where each are may be used at most once during the playing of the game. We prove that a certain problem of game-partition into winning and losing vertices is reducible to strongly connected digraphs. A complete solution for the problem is given in the special case where the digraph is obtained by sticking together the simple cycles at certain vertices.