The Roman domination and domatic numbers of a digraph

Let D be a simple digraph with vertex set V . A Roman dominating function (RDF) on a digraph D is a function f : →{0; 1; 2} satisfying the condition that every vertex v with f(v) = 0 has an in-neighbor u with f(u) = 2. The weight of an RDF f is the value ∑v2V f(v). The Roman domination number of a digraph D is the minimum weight of an RDF on D. A set{ ff1; f2; : : : ; fd} of Roman dominating functions on D with the property that ∑^d_ i=1 fi(v) ≤ 2 for each v Ɛ V , is called a Roman dominating family (of functions) on D. The maximum number of functions in a Roman dominating family on D is the Roman domatic number of D, denoted by d_R(D). In this paper we continue the investigation of the Roman domination number, and we initiate the study of the Roman domatic number in digraphs. We present some bounds for d_R(D). In addition, we determine the Roman domatic number of some digraphs.