Discussiones Mathematicae Graph Theory 28(2) (2008)
A REMARK ON THE (2,2)-DOMINATION NUMBER
Torsten Korneffel, Dirk Meierling and Lutz Volkmann
Lehrstuhl II für Mathematik
RWTH Aachen University, 52056 Aachen, Germany
AbstractA subset D of the vertex set of a graph G is a (k,p)-dominating set if every vertex v ∈ V(G)∖D is within distance k to at least p vertices in D. The parameter γk,p(G) denotes the minimum cardinality of a (k,p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that γk,p(G) ≤ [p/(p+k)]n(G) for any graph G with δk(G) ≥ k+p−1, where the latter means that every vertex is within distance k to at least k+p−1 vertices other than itself. In 2005, Fischermann and Volkmann confirmed this conjecture for all integers k and p for the case that p is a multiple of k. In this paper we show that γ2,2(G) ≤ (n(G)+1)/2 for all connected graphs G and characterize all connected graphs with γ2,2 = (n+1)/2. This means that for k = p = 2 we characterize all connected graphs for which the conjecture is true without the precondition that δ2 ≥ 3.
Keywords: domination, distance domination number, p-domination number.
2000 Mathematics Subject Classification: 05C69.
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Received 2 May 2007
Revised 25 March 2008
Accepted 25 March 2008