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) ∑_xϵN-(v)f(x)≥ k for each vertex vϵV(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)=∑_xϵN(D)f(v). The signed total Italian k -domination number γ𝑘/stIk(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 γ𝑘_st k(D) . In addition, we determine the signed total Italian k -domination number of some classes of digraphs.