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.