Game matching number of graphs

We study a competitive optimization version of α(G), the maximum size of a matching in a graph G. Players alternate adding edges of G to a matching until it becomes a maximal matching. One player (Max) wants that 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 α g(G) and α̂ ′ g(G), respectively. We show that always ∣α′ g(G)− α̂ g(G) ∣∣ ≤ 1. We obtain a sufficient condition for α g(G) = α(G) that is preserved under cartesian product. In general, α g(G) ≥ 2 3α (G), with equality for many split graphs, while α g(G) ≥ 3 4α (G) when G is a forest. Whenever G is a 3regular n-vertex connected graph, α g(G) ≥ n/3, and there are such examples with α g(G) ≤ 7n/18. For an n-vertex path or cycle, the answer is roughly n/7.