Weak Signed Roman Domination in Digraphs

Let D be a finite and simple digraph with vertex set V (D). A weak signed Roman dominating function (WSRDF) on a digraph D is a function f : V (D) -> {-1, 1, 2} satisfying the condition that n-ary sumation Sigma(x is an element of N-[v]) f (x) >= 1 for each v is an element of V (D), where N-[v] consists of v and all vertices of D from which arcs go into v. The weight of a WSRDF f is n-ary sumation Sigma(is an element of V(D)) f (v). The weak signed Roman domination number gamma(wsR)(D) of D is the minimum weight of a WSRDF on D. In this paper we initiate the study of the weak signed Roman domination number of digraphs, and we present different bounds on gamma(wsR)(D). In addition, we determine the weak signed Roman domination number of some classes of digraphs.