# Independent domination number for grid graphs

Let $G_{n,m}$ be the $n \times m$ grid graph, i.e. $G= P_n \Box P_m$, and $T_{n,m}$ the $n\times m$ torus grid graph, i.e. $G= C_n \Box C_m$, where $P_n$ and $C_n$ indicate the path graph of length $n$ and the cycle graph of length $n$, respectively. The independent domination number $i(G)$ is deﬁned to be the minimum cardinality among all maximal independent sets of vertices of $G$.

Is there any explicit formula for $i(G_{n,m})$ and $i(T_{n,m})$? If not, is there a lower bound sharper than the one given by corresponding domination numbers $\gamma(G_{n,m})$ and $\gamma(T_{n,m})$? Or is there any asymptotic results for $i(G_{n,m})/mn$ and $i(T_{n,m})/mn$ as $m,n \to \infty$?