  We study the Maker-Maker version of the domination game introduced in 2018 by
Duch\^ene et al. Given a graph, two players alternately claim vertices. The
first player to claim a dominating set of the graph wins. As the Maker-Breaker
version, this game is PSPACE-complete on split and bipartite graphs. Our main
result is a linear time algorithm to solve this game in forests. We also give a
characterization of the cycles where the first player has a winning strategy.
