Discussiones Mathematicae Graph Theory 27(1) (2007) 51-67
doi: 10.7151/dmgt.1344

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DIGRAPHS WITH ISOMORPHIC UNDERLYING AND DOMINATION GRAPHS: CONNECTED UG^C(D)

Kim A.S. Factor

Marquette University
P.O. Box 1881, Milwaukee, WI 53201-1881, USA
e-mail: kim.factor@marquette.edu

Larry J. Langley

University of the Pacific
3601 Pacific Avenue Stockton, CA 95211, USA
e-mail: llangley@pacific.edu

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