Game matching number of graphs

We study a competitive optimization version of α′(G)α′(G), the maximum size of a matching in a graph GG. Players alternate adding edges of GG to a matching until it becomes a maximal matching. One player (Max) wants the final matching to be large; the other (Min) wants it to be small. The resulting sizes under optimal play when Max or Min starts are denoted αg′(G) and αˆg′(G), respectively. We show that always |αg′(G)−αˆg′(G)|≤1. We obtain a sufficient condition for αg′(G)=α′(G). Always αg′(G)≥23α′(G), with equality for many split graphs, while αg′(G)≥34α′(G) when GG is a forest. Whenever GG is a 33-regular nn-vertex connected graph, αg′(G)≥n/3, and such graphs exist with αg′(G)<7n/18. For an nn-vertex path or cycle, the value is roughly n/7n/7.