UPPER SIGNED k-DOMINATION NUMBER OF DIRECTED GRAPHS

Let k >= 1 be an integer, and let D = (V,A) be a finite simple digraph in which d(D)(-) (v) >= k - 1 for all v is an element of V. A function f : V -> {-1,1} is called a signed k-dominating function (SkDF) if f (N-[v]) >= k for each vertex v is an element of V. An SkDF f of a digraph D is minimal if there is no SkDF g not equal f such that g(v) <= f (v) for each v is an element of V. The maximum values of Sigma(u is an element of v) f(v), taken over all minimal signed k-dominating functions f, is called the upper signed k-domination number Gamma(kS) (D). In this paper, we present a sharp upper bound for Gamma(kS) (D).