Characterisation of forests with trivial game domination numbers

In the Domination Game, two players, the Dominator and Staller, take turns adding vertices of a fixed graph to a set, at each turn increasing the number of vertices dominated by the set, until the final set A∗ dominates the whole graph. The Dominator plays to minimise the size of the set A∗ while the Staller plays to maximise it. A graph is D-trivial if when the Dominator plays first and both players play optimally, the set A∗ is a minimum dominating set of the graph. A graph is S-trivial if the same is true when the Staller plays first. We consider the problem of characterising D-trivial and S-trivial graphs. We give complete characterisations of D-trivial forests and of S-trivial forests. We also show that 2-connected D-trivial graphs cannot have large girth, and conjecture that the same holds without the connectivity condition.