The referred paper is focused on the boundary between the game and complexity theory. It deals with a special type of 2-players game on an acyclic graph, where one of the players chooses a set of vertices and the second player chooses one of the edges starting in one of the chosen vertices. The game is repeated with a limited number of moves. The first player wins when the second player has no other move but the terminal one. The main results characterize the complexity of the considered Q/A games and of some of their modifications. It is also shown that for general strategies the problem is NP complete.