Digraphs with Isomorphic Underlying and Domination Graphs: Connected \u3cem\u3eUG\u3csup\u3ec\u3c/sup\u3e(D)\u3c/em\u3e

The domination graph of a directed graph has an edge between vertices x and y provided either (x,z) or (y,z) is an arc for every vertex z distinct from x and y. We consider directed graphs D for which the domination graph of D is isomorphic to the underlying graph of D. We demonstrate that the complement of the underlying graph must have k connected components isomorphic to complete graphs, paths, or cycles. A complete characterization of directed graphs where k = 1 is presented