Characterizing all nonbipartite well-edge-dominated graphs

Abstract

Given a graph $G$, a set $F$ of edges is an edge dominating set of $G$ if every edge in $G$ is either in $F$ or adjacent to an edge in $F$. A graph $G$ is said to be well-edge-dominated if every minimal edge dominating set has the same cardinality. This definition is the edge version of domination in that a set $D\subseteq V(G)$ is a dominating set if every vertex in $G$ is in $D$ or adjacent to a vertex in $D$ and the domination number ${\hat{I}}^3(G)$ is the minimum cardinality among all dominating sets. In this paper, we complete the characterization of all nonbipartite, well-edge-dominated graphs. In addition, we produce an infinite class of graphs that satisfy the well-known Vizing's conjecture in domination theory that states ${\hat{I}}^3(G\square H)\ge {\hat{I}}^3(G){\hat{I}}^3(H)$ where $G\square H$ is the Cartesian product of $G$ and $H$.