The Domination Game: Proving the 3/5 Conjecture on Isolate-Free Forests

We analyze the domination game, where two players, Dominator and Staller, construct together a dominating set M in a given graph, by alternately selecting vertices into M . Each move must increase the size of the dominated set. The players have opposing goals: Dominator wishes M to be as small as possible, and Staller has the opposite goal. Kinnersley, West and Zamani conjectured in [4] that when both players play optimally on an isolate-free forest, there is a guaranteed upper bound for the size of the dominating set that depends only on the size n of the forest. This bound is 3n/5 when the first player is Dominator, and (3n + 2)/5 when the first player is Staller. The conjecture was proved for specific families of forests in [4] and extended by Bujtás in [2]. Here we prove it for all isolate-free forests, by supplying an algorithm for Dominator that guarantees the desired bound.