On Graphs with Equal Domination and Edge Independence Numbers

Let \(G\) be a simple graph. A set \(D\) of vertices of \(G\) is dominating if every vertex not in \(D\) is adjacent to some vertex in \(D\). A set \(M\) of edges of \(G\) is called independent, or a matching, if no two edges of \(M\) are adjacent in \(G\). The domination number \(\gamma(G)\) is the minimum order of a dominating set in \(G\). The edge independence number \(\alpha_0(G)\) is the maximum size of a matching in \(G\). If \(G\) has no isolated vertices, then the inequality \(\gamma(G) \leq \alpha_0(G)\) holds. In this paper we characterize regular graphs, unicyclic graphs, block graphs, and locally connected graphs for which \(\gamma(G) = \alpha_0(G)\).