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 ∑u∈N[v]f(u)⩾k for every v∈V. An SkDF f of a graph G is minimal if there exists no SkDF g such that g≠f and g(v)⩽f(v) for every v∈V. The maximum of the values of ∑v∈Vf(v), taken over all minimal SkDFs f, is called the upper signed k-domination number ΓkS(G). In this paper, we present a sharp upper bound on this number for a general graph.