Characterization of digraphs with equal domination graphs and underlying graphs (Q2463466)

Let \(D\) be a digraph; the domination graph of \(D\), dom(\(D\)), is the graph associated with \(D\) using the vertex set of \(D\) and \(uv\in E\) [dom(\(D\))] whenever \((u,z)\) or \((v,z)\) are arcs in \(D\) for all vertices \(z\neq u,v\). The underlying graph of a digraph \(D\), denoted by UG(\(D\)), is the graph for which \(D\) is a biorientation. The authors prove that if UG(\(D\))=dom(\(D\)) is a graph on \(n\geq 3\) vertices, then UG(\(D\)) is the join of independent sets, and digraphs whose underlying graphs are identical to their domination graphs are completely characterized. The maximum and minimum number of single arcs in these digraphs is given.