signed total roman k-domination in directed graphs

Let d be a finite and simple digraph with vertex set v(d). a signed total roman k-dominating function (strkdf) on d is a function f:v(d)→{−1,1,2} satisfying the conditions that (i) ∑x∈n−(v)f(x)≥k 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 inner neighbor v for which f(v)=2. the weight of an strkdf f is ω(f)=∑v∈v(d)f(v). the signed total roman k-domination number γ^k str(d) of d is the minimum weight of an strkdf on d. in this paper we initiate the study of the signed total roman k-domination number of digraphs, and we present different bounds on γ^k str(d). in addition, we determine the signed total roman k-domination number of some classes of digraphs. some of our results are extensions of known properties of the signed total roman k-domination number γ^k str(g) of graphs g.