Game-Perfect Semiorientations of Forests

We consider digraph colouring games where two players, Alice and Bob, alternately colour vertices of a given digraph D with a colour from a given colour set in a feasible way. The game ends when such move is not possible any more. Alice wins if every vertex is coloured at the end, otherwise Bob wins. The smallest size of a colour set such that Alice has a winning strategy is the game chromatic number of D. The digraph D is game-perfect if, for every induced subdigraph H of D, the game chromatic number of H equals the size of the largest symmetric clique of H. In the strong game, colouring a vertex is feasible if its colour is different from the colours of its in-neighbours. In the weak game, colouring a vertex is feasible unless it creates a monochromatic directed cycle. There are six variants for each game, which specify the player who begins and whether skipping is allowed for some player. For all six variants of both games, we characterise the class of game-perfect semiorientations of forests by a set of forbidden induced subdigraphs and by an explicit structural description.