Edge-Stable Equimatchable Graphs

TL;DR

This paper characterizes edge-stable equimatchable graphs, which remain equimatchable after any edge removal, providing a recognition algorithm and exploring related stability notions and open questions.

Contribution

It offers a complete characterization of edge-stable equimatchable graphs and an efficient recognition algorithm, advancing understanding of graph stability properties.

Findings

Edge-stable equimatchable graphs are either 2-connected factor-critical or bipartite.

A recognition algorithm with complexity $O( ext{min}(n^{3.376}, n^{1.5}m))$ is provided.

The paper discusses related concepts like edge-critical and vertex-stable equimatchable graphs and open problems.

Abstract

A graph $G$ is \emph{equimatchable} if every maximal matching of $G$ has the same cardinality. We are interested in equimatchable graphs such that the removal of any edge from the graph preserves the equimatchability. We call an equimatchable graph $G$ \emph{edge-stable} if $G\setminus {e}$, that is the graph obtained by the removal of edge $e$ from $G$, is also equimatchable for any $e \in E(G)$. After noticing that edge-stable equimatchable graphs are either 2-connected factor-critical or bipartite, we characterize edge-stable equimatchable graphs. This characterization yields an $O(\min(n^{3.376}, n^{1.5}m))$ time recognition algorithm. Lastly, we introduce and shortly discuss the related notions of edge-critical, vertex-stable and vertex-critical equimatchable graphs. In particular, we emphasize the links between our work and the well-studied notion of shedding vertices, and point out some open questions.