Keeping Avoider's Graph Almost Acyclic.

We consider biased (1 : b) Avoider-Enforcer games in the monotone and strict versions. In particular, we show that Avoider can keep his graph being a forest for every but maybe the last round of the game if b >= 200n ln n. By this we obtain essentially optimal upper bounds on the threshold biases for the non-planarity game, the non-k-colorability game, and the K-t-minor game thus addressing a question and improving the results of Hefetz, Krivelevich, Stojakovie, and Szabo. Moreover, we give a slight improvement for the lower bound in the non-planarity game.