Upper signed k-domination number

Abstract

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 Î 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 Î 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 Î V. The maximum values of åv Î Vf(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).