Critical Equimatchable Graphs

TL;DR

This paper characterizes edge-critical and vertex-critical equimatchable graphs, revealing their structural properties and connections to well-covered graphs, and addresses open questions in graph theory related to matchings and connectivity.

Contribution

It provides a detailed characterization of ECE-graphs and VCE-graphs, including their connectivity and factor-critical properties, and links these to open problems in well-covered graph theory.

Findings

Most ECE-graphs are 2-connected factor-critical graphs.

Bipartite ECE-graphs and even cliques are exceptions.

The paper offers a partial solution to an open question on well-covered graphs.

Abstract

A graph G is equimatchable if every maximal matching of G has the same cardinality. In this paper, we investigate equimatchable graphs such that the removal of any edge harms the equimatchability, called edge-critical equimatchable graphs (ECE-graphs). We show that apart from two simple cases, namely bipartite ECE-graphs and even cliques, all ECE-graphs are 2-connected factor-critical. Accordingly, we give a characterization of factor-critical ECE-graphs with connectivity 2. Our result provides a partial answer to an open question posed by Levit and Mandrescu on the characterization of well-covered graphs with no shedding vertex. We also introduce equimatchable graphs such that the removal of any vertex harms the equimatchability, called vertex-critical equimatchable graphs (VCE-graphs). To conclude, we enlighten the relationship between various subclasses of equimatchable graphs (including ECE-graphs and VCE-graphs) and discuss the properties of factor-critical ECE-graphs with connectivity at least 3.