Signed total Italian domination in digraphs

Let D be a finite and simple digraph with vertex set V (D). A signed total Italian dominating function (STIDF) on a digraph D is a function f : V (D) -> f1; 1; 2g satisfying the conditions that (i) Sigma x is an element of N f(x) >= 1 for each v is an element of V (D), where N(v) consists of all vertices of D from which arcs go into v, and (ii) every vertex u for which f (u) = 1 has an in-neighbor v for which f (v) = 2 or two in-neighbors w and z with f (w) = f(z) = 1. The weight of an STIDF f is Sigma(v is an element of V (D)) f(v). The signed total Italian domination number gamma(stI) (D) of D is the minimum weight of an STIDF on D. In this paper we initiate the study of the signed total Italian domination number of digraphs, and we present different bounds on gamma(stI) (D). In addition, we determine the signed total Italian domination number of some classes of digraphs.