Upper signed k-domination number

Let k 3 1 be an integer and let D = (V, A) be a finite simple digraph in which dD-(v) 3 k - 1 for all v V. A function f V {-1,1} is called a signed k-dominating function (SkDF) if f(N-[v]) 3 k for each vertex v V. An SkDF f of a digraph D is minimal if there is no SkDF g 1 f such that g(v) ￡ f(v) for each v V. The maximum values of v V f(v), taken over all minimal signed k-dominating functions f, is called the upper signed k-domination number GkS(D). In this paper, we present a sharp upper bound for GkS(D).