Signed Domination Number of the Directed Cylinder

In a digraph D = (V (D), A(D)), a two-valued function f : V (D) -> {-1,1} defined on the vertices of D is called a signed dominating function if f(N- [v]) >= 1 for every v in D. The weight of a signed dominating function is f (V(D)) = Sigma(v is an element of v(D)) f (v). The signed domination number gamma(s) (D) is the minimum weight among all signed dominating functions of D. Let P-m x C-n be the Cartesian product of directed path P-m and directed cycle C-n. In this paper, the exact value of gamma(s) (P-m x C-n ) is determined for any positive integers m and n.