Domination game played on trees and spanning subgraphs

The domination game, played on a graph G, was introduced in [3]. Vertices are chosen, one at a time, by two players Dominator and Staller. Each chosen vertex must enlarge the set of vertices of G dominated to that point in the game. Both players use an optimal strategy–Dominator plays so as to end the game as quickly as possible, Staller plays in such a way that the game lasts as many steps as possible. The game domination number γg(G) is the number of vertices chosen when Dominator starts the game and the Staller-start game domination number γ′ g(G) when Staller starts the game. ∗Supported by the Ministry of Science of Slovenia under the grants P1-0297 and J1-2043. The author is also with the Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana. †Research supported by the Hipp Endowed Chair in Mathematics and the Wylie Enrichment Fund of Furman University. Part of the research done during a sabbatical visit at the University of Ljubljana. 1 Pr ep ri n t se ri es , I M FM , I S S N 2 23 220 94 , n o. 1 16 2, S ep te m be r 27 , 2 01 1