Partitioning a graph into a dominating set, a total dominating set, and something else
Michael A. Henning, Christian Löwenstein, Dieter Rautenbach (2010)
Discussiones Mathematicae Graph Theory
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A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, Ars Comb. 89 (2008), 159-162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D∪T necessarily contains all vertices of the graph.