We introduce and study a Maker-Breaker type game in which the issue is to
create or avoid two disjoint dominating sets in graphs without isolated
vertices. We prove that the maker has a winning strategy on all connected
graphs if the game is started by the breaker. This implies the same in the
(2:1) biased game also in the maker-start game. It remains open to
characterize the maker-win graphs in the maker-start non-biased game, and to
analyze the (a:b) biased game for (a:b)=(2:1). For a more restricted
variant of the non-biased game we prove that the maker can win on every graph
without isolated vertices.Comment: 18 page