AI 中文总结
该论文针对$n\times n$方格图的精确距离$k$-控制集，推导了固定$k\geq3$时其最小密度$\delta_k$的上下界，并得到$k=2$时$\delta_2=1/9$的精确值。
AI 中文摘要
设$G_n$为$n\times n$方格图，$k\geq2$，集合$D\subseteq V(G_n)$为精确距离$k$-控制集，当且仅当对每个$v\in V(G_n)\setminus D$，都存在$u\in D$使得$d(u,v)=k$。记$D_{\mathrm{opt}}^{(k)}(G_n)$为这类集合的最小基数，对每个固定$k$，考虑极限$\delta_k=\lim_{n\to\infty} \frac{D_{\mathrm{opt}}^{(k)}(G_n)}{n^2}$。本文证明，对每个固定$k\geq3$，$\frac{1}{4k} \leq \delta_k \leq \frac{k-1}{3k^2-k-1}$；当$k=2$时，可直接得到精确值$\delta_2=1/9$。
英文摘要
Let $G_n$ be the $n\times n$ square grid, and let $k\geq 2$. 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 $D_{\mathrm{opt}}^{(k)}(G_n)$ for the minimum cardinality of such a set. For every fixed $k$, consider the limit $ δ_k= \lim_{n\to\infty} \frac{D_{\mathrm{opt}}^{(k)}(G_n)}{n^2}. $ We prove that, for every fixed \(k\geq 3\), $ \frac{1}{4k} \leq δ_k \leq \frac{k-1}{3k^2-k-1}. $ For $k=2$, the exact value $δ_2=1/9$ follows directly.
