Abstract
A D D 2 -pair of a graph G is a pair ( D , D 2 ) of disjoint sets of vertices of G such that D is a dominating set and D 2 is a 2 -dominating set of G . Although there are infinitely many graphs that do not contain a D D 2 -pair, we show that every graph with minimum degree at least 2 has a D D 2 -pair. We provide a constructive characterization of trees that have a D D 2 -pair and show that K 3 , 3 is the only connected graph with minimum degree at least 3 for which D ∪ D 2 necessarily contains all vertices of the graph.
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