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) → {−1, 1, 2} satisfying the conditions that (i)∑ x∈N−(v) f(x) ≥ 1 for each v ∈ 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 inneighbors w and z with f(w) = f(z) = 1. The weight of an STIDF f is∑ v∈V (D) f(v). The signed total Italian domination number γstI (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 γstI (D). In addition, we determine the signed total Italian domination number of some classes of digraphs.