Domination game: Extremal families of graphs for 3/53/5-conjectures

Two players, Dominator and Staller, alternately choose vertices of a graph GG, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of Dominator is to finish the game as soon as possible, while the aim of Staller is just the opposite. The game domination number γg(G)γg(G) is the number of vertices chosen when Dominator starts the game and both players play optimally. It has been conjectured in Kinnersley et al. (2012) [7] that γg(G)≤3|V(G)|5 holds for an arbitrary graph GG with no isolated vertices, which is in particular open when GG is a forest. In this paper, we present constructions that lead to large families of trees that attain the conjectured 3/53/5-bound. Some of these families can be used to construct graphs with game domination number 3/53/5 of their order by gluing them to an arbitrary graph. All extremal trees on up to 20 vertices were found by a computer. In particular, there are exactly ten trees TT on 20 vertices with γg(T)=12γg(T)=12, all of which belong to the constructed families.