Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21284120190601On independent domination numbers of grid and toroidal grid directed graphs71771384610.22049/cco.2019.26282.1090ENRamyShaheenٍSyrianJournal Article20180622A subset $S$ of vertex set $V(D)$ is an indpendent dominating set of $D$ if $S$ is both an independent and a dominating set of $D$. The indpendent domination number, $i(D)$ is the cardinality of the smallest 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 calculate the independent domination number of the Cartesian product of two directed cycles $C_m$ and $C_n$ for $m, n equiv 0pmod 3$, and $n equiv 0pmod m$. There are many values of $m$ and $n$ such that $C_m Box C_n$ does not have an independent dominating set.http://comb-opt.azaruniv.ac.ir/article_13846_0948ec1c34ebfc23d9e6b9f6dc3f735d.pdf