The signed Roman domatic number of a digraph

A signed Roman dominating function on the digraphD is a function f : V (D) −→ {−1, 1, 2} such that ∑ u∈N−[v] f(u) ≥ 1 for every v ∈ V (D), where N−[v] consists of v and all inner neighbors of v, and every vertex u ∈ V (D) for which f(u) = −1 has an inner neighbor v for which f(v) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman dominating functions on D with the property that ∑d i=1 fi(v) ≤ 1 for each v ∈ V (D), is called a signed Roman dominating family (of functions) on D. The maximum number of functions in a signed Roman dominating family on D is the signed Roman domatic number of D, denoted by dsR(D). In this paper we initiate the study of signed Roman domatic number in digraphs and we present some sharp bounds for dsR(D). In addition, we determine the signed Roman domatic number of some digraphs. Some of our results are extensions of well-known properties of the signed Roman domatic number of graphs.