Characterisation of forests with trivial game domination numbers

Abstract

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.