On independent domination numbers of grid and toroidal grid directed graphs

A subset S of vertex set V (D) is an independent dominating set of a digraph D if S is both an independent and a dominating set of D. The independent domination number i(D) is the minimum cardinality of an independent dominating set of D. In this paper we calculate the independent domination number of the Cartesian product of two directed paths P_m and P_n for arbitraries m and n. Also, we determine the independent domination number of the Cartesian product of two directed cycles C_m and C_n for m,n≡0 (mod 3) and n≡0 (mod m). We note that, there are many values of m and n such that Cm□Cn does not have an independent dominating set.