[k]-Roman Domination in Digraphs

Let D=(V(D),A(D)) be a finite, simple digraph and k a positive integer. A function f:V(D) ? {0, 1, 2, ... ,k+1} is called a [k]-Roman dominating function (for short, [k]-RDF) if f(AN-[v]) = |AN(-)(v)|+k for any vertex v is an element of V(D), where AN(-)(v) = {u? N-(v) : f(u) = 1} and AN(-)[v] = AN(-)(v) ? {v}. The weight of a [k]-RDF f is ?(f) = S(v is an element of V(D))f(v). The minimum weight of any [k]-RDF on D is the [k]-Roman domination number, denoted by ?([kR])(D). For k = 2 and k = 3, we call them the double Roman domination number and the triple Roman domination number, respectively. In this paper, we presented some general bounds and the Nordhaus-Gaddum bound on the [k]-Roman domination number and we also determined the bounds on the [k]-Roman domination number related to other domination parameters, such as domination number and signed domination number. Additionally, we give the exact values of ?([kR])(P-n) and ?([kR])(C-n) for the directed path Pn and directed cycle C-n.