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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.doi:10.7151/dmgt.1344 fatcat:rw22ghmyynccjkv7x6mapbpkiq