BOUNDS ON THE SIGNED ROMAN k-DOMINATION NUMBER OF A DIGRAPH

Let k be a positive integer. A signed Roman k-dominating function (SRkDF) on a digraph D is a function f : V(D) -> {-1, 1, 2} satisfying the conditions that (i) Sigma(x is an element of N)(-[v]) f(x) >= k for each v is an element of V(D), where N-[v] is the closed in-neighborhood of v, and (ii) each vertex u for which f(u) = -1 has an in-neighbor v for which f(v) = 2. The weight of an SRkDF f is Sigma(v is an element of V(D)) f(v). The signed Roman k-domination number gamma(k)(sR)(D) of a digraph D is the minimum weight of an SRkDF on D. We determine the exact values of the signed Roman k-domination number of some special classes of digraphs and establish some bounds on the signed Roman k-domination number of general digraphs. In particular, for an oriented tree T of order n, we show that gamma(2)(sR)(T) >= (n + 3)/2, and we characterize the oriented trees achieving this lower bound.