M. Dorfling, W. Goddard, M.A. Henning and C.M. Mynhardt, Construction of trees and graphs with equal domination parameters, DiscreteMath. 306 (2006) 2647-2654. doi:10.1016/j.disc.2006.04.031[CrossRef]
 S.M. Hedetniemi, S.T. Hedetniemi, R.C. Laskar, L. Markus and P.J. Slater, Disjoint dominating sets in graphs, Proc. ICDM 2006, Ramanujan Mathematics Society Lecture Notes Series 7 (2008) 87-100.
 T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
 M.A. Henning, C. Löwenstein and D. Rautenbach, An independent dominating set in the complement of a minimum dominating set of a tree, Appl. Math. Lett. 23 (2010) 79-81. doi:10.1016/j.aml.2009.08.008[CrossRef] [Web of Science]
 M.A. Henning, C. Löwenstein and D. Rautenbach, Partitioning a graph into a dominating set, a total dominating set, and something else, Discuss. Math. Graph Theory 30 (2010) 563-574. doi:10.7151/dmgt.1514[CrossRef]
 M.A. Henning and J. Southey, A note on graphs with disjoint dominating and total dominating sets, Ars Combin. 89 (2008) 159-162.
 O. Ore, Theory of Graphs: Amer. Math. Soc. Colloq. Publ., 38 (Amer. Math. Soc., Providence, RI, 1962).
 J. Southey and M.A. Henning, Dominating and total dominating partitions in cubic graphs, Central European J. Math. 9(3) (2011) 699-708. doi:10.7151/s11533-011-0014-2[CrossRef]
 J. Southey and M.A. Henning, A characterization of graphs with disjoint dominating and paired-dominating sets, J. Comb. Optim. 22 (2011) 217-234. doi:10.1007/s10878-009-9274-1[CrossRef] [Web of Science]
 B. Zelinka, Total domatic number and degrees of vertices of a graph, Math. Slovaca 39 (1989) 7-11.
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On Graphs with Disjoint Dominating and 2-Dominating Sets
1Department of Mathematics University of Johannesburg Auckland Park, 2006 South Africa
2Department of Mathematics Furman University Greenville, SC, USA
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Citation Information: Discussiones Mathematicae Graph Theory. Volume 33, Issue 1, Pages 139–146, ISSN (Print) 2083-5892, DOI: 10.7151/dmgt.1652, April 2013
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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.