Biased domination games

We consider a biased version of Maker-Breaker domination games, which were recently introduced by Gledel, Ir{š}i{č}, and Klav{ž}ar. Two players, Dominator and Staller, alternatingly claim vertices of a graph $G$ where Dominator is allowed to claim up to $b$ vertices in every round and she wins if and only if she occupies all vertices of a dominating set of $G$. For this game, we prove a full characterization of all trees on which Dominator has a winning strategy. For the number of rounds which Dominator needs to win, we give exact results when played on powers of paths or cycles, and for all trees we provide bounds which are optimal up to a constant factor not depending on $b$. Furthermore, we discuss general minimum degree conditions and study how many vertices can still be dominated by Dominator even when Staller has a winning strategy.