On well-edge-dominated graphs

TL;DR

This paper characterizes well-edge-dominated graphs, showing that certain classes are limited to specific small graphs and providing characterizations for split and Cartesian product graphs.

Contribution

It provides a complete classification of connected, triangle-free, nonbipartite, well-edge-dominated graphs and characterizes well-edge-dominated split and Cartesian product graphs.

Findings

Connected, triangle-free, nonbipartite, well-edge-dominated graphs are limited to three specific graphs.

Connected Cartesian product graphs are well-edge-dominated only if they are $K_2 \Box K_2$.

Characterizations of well-edge-dominated split graphs are provided.

Abstract

A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum. It is known that every well-edge-dominated graph $G$ is also equimatchable, meaning that every maximal matching in $G$ is maximum. In this paper, we show that if $G$ is a connected, triangle-free, nonbipartite, well-edge-dominated graph, then $G$ is one of three graphs. We also characterize the well-edge-dominated split graphs and Cartesian products. In particular, we show that a connected Cartesian product $G\Box H$ is well-edge-dominated, where $G$ and $H$ have order at least $2$, if and only if $G\Box H = K_2 \Box K_2$.