Games of no return a

Let D = (V,A) be a finite directed graph (digraph) each vertex v ∈ V of which is interpreted as a position and each arc a = (v, v′) ∈ A as a possible move from position v to v′. Two players, 1 and 2, take turns moving a token from a given initial position v0. The game ends as soon as the token returns to a position, where it has already been. By definition, the player who made the last move loses, while the opponent wins (in the standard version, and vice versa in the misere version). The defined games of no return generalize classical combinatorial (NIM-type) games. Indeed, every such game (D, v0) on an acyclic digraph D turns into a game of no return (D′, v0) after adding a loop to every terminal vertex of D. In this case, D has a unique kernel K ⊆ V and player 2 wins iff v0 ∈ K. The kernel can be obtained in linear time by the well-known von Neumann algorithm. Interestingly, a no return games (D, v0) can be solved in polynomial time also in case when digraph D = (V,A) is symmetric, that is, a = (v, v′) ∈ A iff a′ = (v′, v) ∈ A. Let us replace every such pair of arcs by a non-directed edge e = (v, v′) and denote by G the obtained non-directed graph. We show that player 1 wins in the symmetric game (D, v0) iff every maximum matching of G contains v0. Moreover, a move (v0, v) is winning iff edge (v0, v) belongs to such a maximum matching. Both conditions can be verified in polynomial time, due to the polynomial algorithm for finding a maximum matching of a graph given by the first author in 1965. We suggest a polynomial algorithm for solving games for a slightly larger class of digraphs, which contains both acyclic and symmetric ones. Let us note that these games (as well as the symmetric and combinatorial games) can be solved in the positional strategies, that is, an optimal move (v, v′) in any position v depends only on v, not on the preceding positions and moves. Yet, for general digraphs, it is not always the case and a winning strategy might require an exponential description. We show that problem Q of solving a no return game is both NPand coNP-hard. Moreover, Q ∈ NP iff Q ∈ coNP . Thus, it is unlikely that Q belongs to NP or coNP. Indeed, then Q would be both NPand coNP-complete, implying NP = coNP . In fact, the same construction proves NP-hardness of the four types of games, in which a position or move should or should not be repeated; the games of (no) return and of (no) repetition. Yet, for the acyclic and symmetric digraphs, problem Q is polynomial in all four cases.