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.
@article{bwmeta1.element.doi-10_7151_dmgt_1652,
author = {Michael A. Henning and Douglas F. Rall},
title = {On Graphs with Disjoint Dominating and 2-Dominating Sets},
journal = {Discussiones Mathematicae Graph Theory},
volume = {33},
year = {2013},
pages = {139-146},
zbl = {1291.05144},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1652}
}
Michael A. Henning; Douglas F. Rall. On Graphs with Disjoint Dominating and 2-Dominating Sets. Discussiones Mathematicae Graph Theory, Tome 33 (2013) pp. 139-146. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1652/
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