Signed Roman k-domination in Digraphs

Let D be a finite and simple digraph with vertex set V (D) and arc set A(D). A signed Roman dominating function (SRDF) on the digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑ x∈N−[v] f(x) ≥ 1 for each v ∈ V (D), where N −[v] consists of v and all inner neighbors of v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The weight of an SRDF f is w(f) = ∑ v∈V (D) f(v). The signed Roman domination number γsR(D) of D is the minimum weight of an SRDF on D. In this paper we initiate the study of the signed Roman domination number of digraphs, and we present different bounds on γsR(D). In addition, we determine the signed Roman domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the signed Roman domination number γsR(G) of graphs G.