Signed total Italian k-domination in digraphs

Let k > 1 be an integer, and let D be a finite and simple digraph with vertex set V (D). A signed total Italian k-dominating function (STIkDF) on a digraph D is a function f : V (D)-+ {-1, 1,2} satisfying the conditions that (i) Ex is an element of N-(v) f (x) > k for each vertex v E V (D), where N-(v) consists of all vertices of D from which arcs go into v, and (ii) each vertex u with 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 STIkDF f is omega(f) = sigma v is an element of V (D) f (v). The signed total Italian k-domination number gamma kstI(D) of D is the minimum weight of an STIkDF on D. In this paper we initiate the study of the signed total Italian k-domination number of digraphs, and we present different bounds on gamma kstI (D). In addition, we determine the signed total Italian k-domination number of some classes of digraphs.