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 ⊆ 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 γ(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 γ(G□H) ≥ γ(G)γ(H) where G□H is the Cartesian product of G and H.