The upper bound on k-tuple domination numbers of graphs

Authors:
Gerard J. Chang
Affiliations:
Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan and Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan and National Center for Theore ...
Venue:
European Journal of Combinatorics
Year:
2008

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Abstract

In a graph G, a vertex is said to dominate itself and all vertices adjacent to it. For a positive integer k, the k-tuple domination number @c"x"k(G) of G is the minimum size of a subset D of V(G) such that every vertex in G is dominated by at least k vertices in D. To generalize/improve known upper bounds for the k-tuple domination number, this paper establishes that for any positive integer k and any graph G of n vertices and minimum degree @d, @c"x"k(G)@?ln(@d-k+2)+lnd@?"k"-"1+1@d-k+2n, where d@?"m=1n@?"i"="1^nd"i+1m with d"i the degree of the ith vertex of G.