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2, girth 
. This result was recently improved by Volkmann [Upper bounds on the domination number of a graph in terms of diameter and girth, J. Combin. Math. Combin. Comput. 52 (2005) 131–141; An upper bound for the domination number of a graph in terms of order and girth, J. Combin. Math. Combin. Comput. 54 (2005) 195–212] who for 
doi:10.1016/j.aml.2006.03.006 
Copyright © 2006 Elsevier Ltd All rights reserved.
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Received 3 February 2006;
accepted 6 March 2006.
Available online 4 May 2006.
Abstract
Let G=(V,E) be a graph, and let NG(v) and dG(v) denote the neighbourhood and degree of a vertex v
V in G, respectively. The minimum cardinality of a set D
V with |NG(v)∩D|≥k for all vV
D is the k-domination number γk(G) of G. Similarly, the minimum cardinality of a set DV with |(NG(v)
{v})∩D|≥k for all vV is the k-tuple domination number γ×k(G) of G.
Let G be a graph of order n and minimum degree δ and let . We prove that if
, then
Furthermore, we prove that if δ≥2, then
which generalizes a recent result of J. Harant and M. Henning.
