Discussiones Mathematicae Graph Theory 27(1) (2007) 93-103
doi: 10.7151/dmgt.1347

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CHARACTERIZATION OF BLOCK GRAPHS WITH EQUAL 2-DOMINATION NUMBER AND DOMINATION NUMBER PLUS ONE

Adriana Hansberg  and  Lutz Volkmann

Lehrstuhl II für Mathematik
RWTH Aachen University
52056 Aachen, Germany
e-mail: hansberg@math2.rwth-aachen.de
e-mail: volkm@math2.rwth-aachen.de

Abstract

Let G be a simple graph, and let p be a positive integer. A subset D ⊆ V(G) is a p-dominating set of the graph G, if every vertex v ∈ V(G)−D is adjacent with at least p vertices of D. The p-domination number γp(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ1(G) is the usual domination number γ(G).

If G is a nontrivial connected block graph, then we show that γ2(G) ≥ γ(G)+1, and we characterize all connected block graphs with γ2(G) = γ(G)+1. Our results generalize those of Volkmann [12] for trees.

Keywords: domination, 2-domination, multiple domination, block graph.

2000 Mathematics Subject Classification: 05C69.

References

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Received 7 December 2005
Revised 18 October 2006