Discussiones Mathematicae Graph Theory 27(1) (2007)
51-67
DOI: https://doi.org/10.7151/dmgt.1344
DIGRAPHS WITH ISOMORPHIC UNDERLYING AND DOMINATION GRAPHS: CONNECTED UG^C(D)
Kim A.S. Factor
Marquette University | Larry J. Langley
University of the Pacific |
Dedicated to the memory of Kenneth P. Bogart
Abstract
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.
2000 Mathematics Subject Classification: 05C69.
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Received 21 September 2005
Revised 24 June 2006
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