Geodesic
On Graphs with Disjoint Dominating and 2-Dominating Sets
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 139-146

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A DD2-pair of a graph G is a pair (D,D2) of disjoint sets of vertices of G such that D is a dominating set and D2 is a 2-dominating set of G. Although there are infinitely many graphs that do not contain a DD2-pair, we show that every graph with minimum degree at least two has a DD2-pair. We provide a constructive characterization of trees that have a DD2-pair and show that K3,3 is the only connected graph with minimum degree at least three for which D ∪ D2 necessarily contains all vertices of the graph.
Keywords: domination, 2-domination, vertex partition 
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Henning, Michael A.; Rall, Douglas F. On Graphs with Disjoint Dominating and 2-Dominating Sets. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 139-146. http://geodesic.mathdoc.fr/item/DMGT_2013_33_1_a11/