The matcher game played in graphs

We study a game played on a graph by two players, named Maximizer and Minimizer. Each round two new vertices are chosen; first Maximizer chooses a vertex u that has at least one unchosen neighbor and then Minimizer chooses a neighbor of u. This process eventually produces a maximal matching of the graph. Maximizer tries to maximize the number of edges chosen, while Minimizer tries to minimize it. The matcher number α_g^′(G) of a graph G is the number of edges chosen when both players play optimally. In this paper it is proved that α_g^′(G)≥[Formula presented]α^′(G), where α^′(G) denotes the matching number of graph G, and this bound is tight. Further, if G is bipartite, then α_g^′(G)=α^′(G). We also provide some results on graphs of large odd girth and on dense graphs.