Edge-Critical Equimatchable Graphs

• Edge-critical equimatchable graphs are defined as finite, simple graphs where every maximal matching is equimatchable and removal of any edge disrupts this property, linking them to well-covered line graphs.
• The classification divides ECE-graphs into distinct types—factor-critical, bipartite, and even cliques—providing a clear structural framework to address open problems in graph theory.
• Detailed analysis of connectivity, including specific cases for 2-connected and higher, yields actionable insights into the combinatorial properties and resilience of these graphs.

An edge-critical equimatchable graph (ECE-graph) is a finite, simple graph in which every maximal matching has the same cardinality (i.e., the graph is equimatchable), but the removal of any edge destroys this property. The study of ECE-graphs arises from the combinatorial theory of matchings and is intimately connected to the structure theory of well-covered line graphs and factor-criticality. Beyond their intrinsic structural interest, ECE-graphs are central to the resolution of open problems on well-covered graphs without shedding vertices, notably the Levit–Mandrescu problem (Deniz et al., 2022).

1. Foundational Definitions and Properties

A matching in a graph $G$ is a set of edges no two of which share an endvertex. The matching is maximal if it is not properly contained in any other matching. A graph is equimatchable if every maximal matching has the same cardinality, which coincides with its matching number $\nu(G)$: for every maximal matching $M\subseteq E(G)$, $|M|=\nu(G)$. An edge $e\in E(G)$ is called critical if $G-e$ is not equimatchable. A graph is edge-critical equimatchable (ECE-graph) if every edge of $G$ is critical.

Equimatchable graphs are equivalent to well-covered line graphs, providing a direct link to well-coveredness: for such $G$ (with no $K_2$-components), $L(G)$ is well-covered and has no shedding vertex. Consequently, classifying ECE-graphs provides large, explicit classes of well-covered graphs lacking any shedding vertex—a key unresolved topic.

2. Structure and Global Classification

Every ECE-graph is necessarily 2-connected (Deniz–Ekim Lemma 3.6). They are classified into the following exclusive types (Deniz–Ekim Theorem 3.7):

• Type (i): Factor-Critical—$\nu(G)$0 is factor-critical and 2-connected.
• Type (ii): Bipartite—$\nu(G)$1 is bipartite and 2-connected.
• Type (iii): Even Clique—$\nu(G)$2 for some $\nu(G)$3.

With the exception of bipartite ECE-graphs and even cliques, all ECE-graphs are factor-critical. The characterization of bipartite ECE-graphs is as follows: let $\nu(G)$4 be a connected bipartite graph with $\nu(G)$5, $\nu(G)$6; then $\nu(G)$7 is ECE if and only if for every $\nu(G)$8 and every nonempty $\nu(G)$9, $M\subseteq E(G)$0, with equality only when $M\subseteq E(G)$1 (Deniz et al., 2022).

3. Factor-Critical ECE-Graphs with Connectivity Two

If $M\subseteq E(G)$2 is a factor-critical ECE-graph with connectivity 2, its structure admits a complete classification via Favaron’s theorem and a precise case analysis. Given such a graph with a 2-cut $M\subseteq E(G)$3, $M\subseteq E(G)$4 has exactly two components $M\subseteq E(G)$5 and $M\subseteq E(G)$6 with the following isomorphism types:

• $M\subseteq E(G)$7 or $M\subseteq E(G)$8
• $M\subseteq E(G)$9 or $|M|=\nu(G)$0

Here, 2-cut vertices are adjacent to designated $|M|=\nu(G)$1 and $|M|=\nu(G)$2. Central to the classification is the notion of "partial-completeness" to a component: for $|M|=\nu(G)$3 partially-complete to $|M|=\nu(G)$4, there exists a partition such that $|M|=\nu(G)$5 is complete to one part and nonadjacent to the other. The complete family $|M|=\nu(G)$6 of 2-connected, factor-critical ECE-graphs consists of five types:

Type	Structure of $|M|=\nu(G)$7 and $|M|=\nu(G)$8	Notable adjacency
I	$|M|=\nu(G)$9, $e\in E(G)$0	$e\in E(G)$1 complete to $e\in E(G)$2; partial to $e\in E(G)$3
II	$e\in E(G)$4, $e\in E(G)$5	Each $e\in E(G)$6 complete to a $e\in E(G)$7-part of $e\in E(G)$8; partial to $e\in E(G)$9
III	$G-e$0, $G-e$1	$G-e$2 complete to $G-e$3; partial to large part of $G-e$4
IV	$G-e$5, $G-e$6	Each $G-e$7 complete to part of $G-e$8; partial to large part of $G-e$9
V	$G$0, $G$1	Special $G$2; complex local join conditions

A factor-critical graph of connectivity 2 is ECE if and only if it belongs to this family (Deniz–Ekim Theorem 4.11) (Deniz et al., 2022).

4. Higher Connectivity ECE-Graphs

The structural understanding of factor-critical ECE-graphs with connectivity at least three is partial. For 3-connected graphs where deletion of a 3-cut leaves two components, each of size at least three, the components are complete, and edges to the cut set induce three additional types (termed VI–VIII). The case where one component has size two is also addressed (Deniz et al., 2022).

For connectivity $G$3 and $G$4 with minimum degree $G$5, factor-critical ECE-graphs are those with independence number two and no dominating edge. Equivalently, such a $G$6 is an ECE-graph if and only if its complement $G$7 is a maximal triangle-free graph of diameter 2 (Eiben–Kotrbcík, Lemma 4.2 in (Deniz et al., 2022)). Cases with $G$8 or small order ($G$9) are left open.

5. Connections to Well-Covered Graphs and Open Problems

A well-covered graph is 1-well-covered if every vertex is a shedding vertex. The complete structural description of well-covered graphs without any shedding vertex—posed by Levit and Mandrescu—is a recognized open problem. Since an ECE-graph $G$0 with no $G$1-component satisfies that $G$2 is well-covered and has no shedding vertex, a full classification of ECE-graphs would resolve the problem for the class of line graphs.

These results, especially the explicit 2-connected, factor-critical, and bipartite ECE-graph classifications, provide the first full families of well-covered line graphs without shedding vertices and furnish explicit instances for further investigation (Deniz et al., 2022).

6. Relationship with Related Equimatchable Graph Classes

ECE-graphs are one subclass within the broader category of equimatchable graphs. There are also vertex-critical equimatchable graphs (VCE-graphs), where removal of any vertex destroys equimatchability. The relationships among these subclasses, and their further characterization, are discussed in the context of equimatchable, well-covered, and factor-critical graphs, illuminating how edge-criticality imposes strong structural restrictions compared to the general equimatchable case.

The ongoing development of ECE-graph theory continues to impact the structural theory of matchings, the characterization of well-covered line graphs, and the combinatorics of factor-criticality, with multiple open directions related to connectivity and extremal structure still to be resolved (Deniz et al., 2022).