Exact-Distance Domination in Grid Graphs

Let $G_n$ be the $n\times n$ square grid, and let $k\ge 2$. A set $D\subseteq V(G_n)$ is an \emph{exact-distance $k$-dominating set} if every vertex $v\in V(G_n)\setminus D$ has a vertex $u\in D$ with $d(u,v)=k$. We write $D_{\mathrm{o}\mathrm{p}\mathrm{t}}^{(k)}(G_n)$ for the minimum cardinality of such a set. For every fixed $k$, consider the limit ${\delta }_k={\lim }_{n\to \mathrm{\infty }}\frac{D_{\mathrm{o}\mathrm{p}\mathrm{t}}^{(k)}(G_n)}{n^2}\mathrm{.}$ We prove that, for every fixed $k\ge 3$, $\frac{1}{4k}\le {\delta }_k\le \frac{k-1}{3k^2-k-1}\mathrm{.}$ For $k=2$, the exact value ${\delta }_2=1\mathrm{/}9$ follows directly.