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Description:
A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, {\it Ars Comb.} {\bf 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 $D\cup T$ necessarily contains all vertices of the graph.