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 $$A_*$$ A ∗ dominates the whole graph. The Dominator plays to minimise the size of the set $$A_*$$ A ∗ while the Staller plays to maximise it. A graph is $$D$$ D -trivial if when the Dominator plays first and both players play optimally, the set $$A_*$$ A ∗ is a minimum dominating set of the graph. A graph is $$S$$ S -trivial if the same is true when the Staller plays first. We consider the problem of characterising $$D$$ D -trivial and $$S$$ S -trivial graphs. We give complete characterisations of $$D$$ D -trivial forests and of $$S$$ S -trivial forests. We also show that $$2$$ 2 -connected $$D$$ D -trivial graphs cannot have large girth, and conjecture that the same holds without the connectivity condition.