Title: On graphs with disjoint dominating and 2-dominating sets
Abstract: A DD 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 DD 2 -pair, we show that every graph with minimum degree at least two has a DD 2 -pair.We provide a constructive characterization of trees that have a DD 2 -pair and show that K 3,3 is the only connected graph with minimum degree at least three for which D ∪ D 2 necessarily contains all vertices of the graph.