EXTREMAL PROBLEMS FOR GAME DOMINATION NUMBER

In the domination game on a graph G, two players called Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices dominated; the game ends when the chosen set becomes a dominating set of G. Dominator aims to minimize the size of the resulting dominating set, while Staller aims to maximize it. When both players play optimally, the size of the dominating set produced is the game domination number of G, denoted by gamma(g)(G) when Dominator plays first and by gamma(g)' (G) when Staller plays first. We prove that gamma(g)(G) <= 7n/11 when G is an isolate-free n-vertex forest and that gamma(g)(G) = <= inverted right perpendicular7n/10inverted left perpendicular for any isolate-free n-vertex graph. In both cases we conjecture that gamma(g)(G) <= 3n/5 and prove it when G is a forest of nontrivial caterpillars. We also resolve conjectures of Bresar, Klavzar, and Rall by showing that always gamma(g)' (G) <= gamma(g)(G)+1, that for k >= 2 there are graphs G satisfying gamma(g)(G) = 2k and gamma(g)' (G) = 2k - 1, and that gamma(g)' (G) >= gamma(g)(G) when G is a forest. Our results follow from fundamental lemmas about the domination game that simplify its analysis and may be useful in future research.