Domination Game and an Imagination Strategy

The domination game played on a graph G consists of two players, Dominator and Staller who alternate taking turns choosing a vertex from G such that whenever a vertex is chosen by either player, at least one additional vertex is dominated. Dominator wishes to dominate the graph in as few steps as possible and Staller wishes to delay the process as much 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. An imagination strategy is developed as a general tool for proving results on the domination game. We show that for any graph G, γ(G) ≤ γg(G) ≤ 2γ(G) − 1, and that all possible values can be realized. It is proved that for any graph G, γg(G) − 1 ≤ γ′ g(G) ≤ γg(G) + 2, and that most of the possibilities for mutual values of γg(G) and γ′ g(G) can be realized. A connection with Vizing’s conjecture is established, and a lower bound on the ∗The paper was initiated during the visit of the third author to the IMFM and completed during the visit of the first two authors at Furman University supported in part by the Slovenia-USA bilateral grant BI-US/08-10-004 and the Wylie Enrichment Fund of Furman University. †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 Ljubl-