Characterizing all $K_4$-free well-edge-dominated graphs of girth 3
Sarah E. Anderson, Kirsti Kuenzel
TL;DR
This work advances the structural classification of well-edge-dominated graphs by focusing on the $K_4$-free case with a $3$-cycle. The authors build three infinite families, namely Propellers $\mathcal{P}$, Windmills $\mathcal{W}$, and a composite class $\mathcal{G}$, and prove that any connected, nonbipartite, $K_4$-free well-edge-dominated graph of girth $3$ must lie in $\mathcal{G}$ or in a small finite set $\{\mathcal{DH}, F_5, Cr, W_1, W_2, W_3\}$. This dichotomy is supported by a constructive framework showing how $\mathcal{G}$ is closed under specific identifications of well-edge-dominated components, and by computational checks that confirm the complete lists up to order $8$. The results complement existing classifications for girth $\ge 4$ and for graphs with exactly one triangle, moving toward a full description of non-bipartite well-edge-dominated graphs with forbidden $K_4$ subgraphs. The findings contribute to understanding edge domination and equimatchability in small-girth graphs and provide a foundation for extending to graphs containing induced $K_4$-subgraphs.
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.