Geodesic
Partitioning a graph into a dominating set, a total dominating set, and something else
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 563-574

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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.
Keywords: domination, total domination, domatic number, vertex partition, Petersen graph 
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     title = {Partitioning a graph into a dominating set, a total dominating set, and something else},
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Henning, Michael; Löwenstein, Christian; Rautenbach, Dieter. Partitioning a graph into a dominating set, a total dominating set, and something else. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 563-574. http://geodesic.mathdoc.fr/item/DMGT_2010_30_4_a3/