Discussiones Mathematicae Graph Theory 33(1) (2013) 139-146
doi: 10.7151/dmgt.1652

This paper is dedicated to Mietek Borowiecki on the occasion of his 70th
birthday. We thank him for his warmth and kindness to us, for his inspiration,
and for his dedication to graph theory.

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On graphs with disjoint dominating and 2-dominating sets

Michael A. Henning

Department of Mathematics
University of Johannesburg
Auckland Park, 2006 South Africa

Douglas F. Rall

Department of Mathematics
Furman University
Greenville, SC, USA

Abstract

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

2010 Mathematics Subject Classification: 05C69.

References

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Received 28 February 2012
Revised 2 October 2012
Accepted 31 October 2012