Geodesic
Digraphs with isomorphic underlying and domination graphs: connected $UG^c(d)$
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 51-67.

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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.
Keywords: domination graph, domination, graph isomorphism, underlying graph 
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Factor, Kim; Langley, Larry. Digraphs with isomorphic underlying and domination graphs: connected $UG^c(d)$. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 51-67. http://geodesic.mathdoc.fr/item/DMGT_2007_27_1_a5/

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