SIGNED DOMINATION NUMBER OF COMPLETE DIRECTED GRID GRAPH

Let Dbe a finite simple directed graphwith vertex set V(D)and arc set A(D). Letfbe a mapping of the vertex set V(D) into set {-1, 1}. If 1])[(vNfDfor each vertex vV(D), then fis called a signed dominating function on D. The weight ω(f )of fis defined by Vvvf)(. The signed domination number of a digraph D is s(D) = min{ω(f ): fis a signed dominating functionof D}.Let Pm×Pndenote the cartesian product of directed paths of length m and n. In this paper, we determine the exact values of s(Pm×Pn) for m = 8, 9, 10 and arbitrary n.Also, we give a lower bound of s(Pm×Pn)