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.