Game domination number

Abstract

The game domination number of a (simple, undirected) graph is defined by the following game. Two players, $\text{A}$ and $\text{D}$, orient the edges of the graph alternately until all edges are oriented. Player $\text{D}$ starts the game, and his goal is to decrease the domination number of the resulting digraph, while $\text{A}$ is trying to increase it. The game domination number of the graph $\text{G}$, denoted by $\text{γ}{}_{\mathrm{<mtext>g</mtext>}}\text{(G)}$, is the domination number of the directed graph resulting from this game. This is well defined if we suppose that both players follow their optimal strategies. We determine the game domination number for several classes of graphs and provide general inequalities relating it to other graph parameters.