Upper signed k-domination in a general graph

Let k be a positive integer, and let G = (V, E) be a graph with minimum degree at least k - 1. A function f : V -> {-1, 1} is said to be a signed k-dominating function (SkDF) if Sigma(u is an element of N[v]) f (u) >= k for every v is an element of V. An SkDF f of a graph G is minimal if there exists no SkDF g such that g not equal f and g(v) <= f (v) for every v is an element of V. The maximum of the values of Sigma(v is an element of V) f (v), taken over all minimal SkDFs f, is called the upper signed k-domination number Gamma(kS)(G). In this paper, we present a sharp upper bound on this number for a general graph. Crown Copyright (C) 2010 Published by Elsevier B.V. All rights reserved.