Deciding the winner in k rounds for DISJOINT ARROWS, a new combinatorial partizan game

We consider DISJOINT ARROWS, a new bounded-length two-player partizan combinatorial game. In this game the two players, Alice and Bob, alternate in choosing vertices on a directed graph and no player is allowed to select a vertex previously selected. At each round Alice selects a vertex u and Bob has to reply choosing a new vertex in the out-neighborhood of u. The first player who cannot move loses. We prove that deciding whether Bob can endure k rounds when k is part of the input is a PSPACE-complete problem. Moreover we prove that the parameterized version of the problem is AW[*]-complete. Thus we provide a new member for the small set of problems known complete for the class AW[*].