Game matching number of graphs

TL;DR

This paper investigates a competitive game on graphs where two players alternately add edges to form a maximal matching, analyzing the maximum and minimum sizes achievable under optimal play and establishing bounds and conditions related to the graph's structure.

Contribution

It introduces a new game-theoretic framework for graph matchings, providing bounds, conditions, and exact values for various classes of graphs, including paths, cycles, and regular graphs.

Findings

The difference between Max and Min matchings is at most 1.

A sufficient condition for Max(G) to equal the maximum matching size is identified.

Bounds are established for Max(G) in terms of the graph's maximum matching and specific graph classes.

Abstract

We study a competitive optimization version of $\alpha'(G)$, 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 $\Max(G)$ and $\Min(G)$, respectively. We show that always $|\Max(G)-\Min(G)|\le 1$. We obtain a sufficient condition for $\Max(G)=\alpha'(G)$ that is preserved under cartesian product. In general, $\Max(G)\ge \frac23\alpha'(G)$, with equality for many split graphs, while $\Max(G)\ge\frac34\alpha'(G)$ when $G$ is a forest. Whenever $G$ is a 3-regular $n$-vertex connected graph, $\Max(G) \ge n/3$, and there are such examples with $\Max(G)\le 7n/18$. For an $n$-vertex path or cycle, the answer is roughly $n/7$.