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

Two players, Dominator and Staller, alternately choose vertices of a graph G, 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 gamma(g) (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 gamma(g) (G) <= 3 vertical bar V(G)vertical bar/5 holds for an arbitrary graph G with no isolated vertices, which is in particular open when G is a forest. In this paper, we present constructions that lead to large families of trees that attain the conjectured 3/5-bound. Some of these families can be used to construct graphs with game domination number 3/5 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 T on 20 vertices with gamma(g) (T) = 12, all of which belong to the constructed families. (C) 2013 Elsevier B.V. All rights reserved.