Abstract
The domination game is played on a graph G by Dominator and Staller. The game domination number ?(G) of G is the number of moves played when Dominator starts and both players play optimally. Similarly, ?g (G) is the number of moves played when Staller starts. Graphs G with ?(G) = 2, graphs with ?g(G) = 2, as well as graphs extremal with respect to the diameter among these graphs are characterized. In particular, ?g (G) = 2 and diam(G) = 3 hold for a graph G if and only if G is a so-called gamburger. Graphs G with ?(G) = 3 and diam(G) = 6, as well as graphs G with ?g(G) = 3 and diam(G) = 5 are also characterized.
Highlights
The domination game is played on an arbitrary graph G by Dominator and Staller
The game ends when no move is possible and the score of the game is the total number of vertices chosen
By D-game we mean a game in which Dominator has the first move and by S-game a game started by Staller
Summary
The domination game is played on an arbitrary graph G by Dominator and Staller. The two players take turns choosing a vertex from G such that at least one previously undominated vertex becomes dominated. Assuming that both players play optimally, the game domination number γg(G) of a graph G denotes the score of D-game played on G. The Staller-start game domination number γg′ (G) is defined as the score of optimal S-game. We introduce notations needed, recall some results, and bound the diameter of a graph from above in terms of the game domination number (see [3, Corollary 4.1] for a closely related result).
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