Characterizing all $K_4$-free well-edge-dominated graphs of girth 3

TL;DR

This paper characterizes all $K_4$-free, well-edge-dominated graphs containing a triangle, extending understanding of edge domination properties in graphs with specific girth and clique constraints.

Contribution

It provides a complete characterization of $K_4$-free, well-edge-dominated graphs with triangles, filling a gap in the classification of such graphs.

Findings

Identified all $K_4$-free, well-edge-dominated graphs with triangles.

Extended previous classifications to include girth 3 graphs.

Provided structural insights into edge domination in constrained graphs.

Abstract

Given a graph $G$, a set $F$ of edges is an edge dominating set if all edges in $G$ are either in $F$ or adjacent to an edge in $F$. $G$ is said to be well-edge-dominated if every minimal edge dominating set is also minimum. In 2022, it was proven that there are precisely three nonbipartite, well-edge-dominated graphs with girth at least four. Then in 2025, a characterization of all well-edge-dominated graphs containing exactly one triangle was found. In this paper, we characterize all well-edge-dominated graphs that contain a triangle and yet are $K_4$-free.