On two extensions of equimatchable graphs

TL;DR

This paper introduces two new parameters to measure how far a graph is from being equimatchable, providing complexity results, characterizations, and bounds for these parameters.

Contribution

It defines the matching gap and equimatchability defect, analyzes their computational complexity, and characterizes graphs with specific properties related to these parameters.

Findings

Characterization of graphs with unit matching gap

Exact equimatchability defect values for cycles

Bounds for matching gap and equimatchability defect

Abstract

A graph is said to be equimatchable if all its maximal matchings are of the same size. In this work we introduce two extensions of the property of equimatchability by defining two new graph parameters that measure how far a graph is from being equimatchable. The first one, called the matching gap, measures the difference between the sizes of a maximum matching and a minimum maximal matching. The second extension is obtained by introducing the concept of equimatchable sets; a set of vertices in a graph $G$ is said to be equimatchable if all maximal matchings of $G$ saturating the set are of the same size. Noting that $G$ is equimatchable if and only if the empty set is equimatchable, we study the equimatchability defect of the graph, defined as the minimum size of an equimatchable set in it. We develop several inapproximability and parameterized complexity results and algorithms regarding the computation of these two parameters, a characterization of graphs of unit matching gap, exact values of the equimatchability defect of cycles, and sharp bounds for both parameters.