The signed total Roman domatic number of a digraph

Let D be a finite simple digraph with vertex set V (D). A signed total Roman dominating function (STRDF) on a digraph D is a function f : V (D) -> {- 1, 1, 2} such that (i) Sigma(u is an element of N)-((v)) f(u) >= 1 for every v is an element of V (D), where N- (v) consists of all inner neighbors of v, and (ii) every vertex u is an element of V (D) for which f(u) = -1 has an inner neighbor v for which f(v) = 2. The weight of an STRDF f is omega(f) = Sigma(v is an element of V) ((D)) f(v). The signed total Roman domination number gamma(stR)(D) of D is the minimum weight of an STRDF on D. A set {f(1), f(2),..., f(d)} of distinct STRDFs on D with the property that Sigma(d)(i=1) f(i)(v) <= 1 for each v is an element of V (D) is called a signed total Roman dominating family (STRD family) (of functions) on D. The maximum number of functions in an STRD family on D is the signed total Roman domatic number of D, denoted by d(stR)(D). In this paper, we initiate the study of signed total Roman domatic number in digraphs and we present some sharp bounds for d(stR)(D). In addition, we determine the signed total Roman domatic number of some classes of digraphs.