Exact-Distance Domination in Grid Graphs

2026-07-31

AI Summary

This study investigates the asymptotic density $\delta_k$ of minimum exact distance-$k$ dominating sets in $n \times n$ grid graphs for $k \geq 2$, where every non-dominated vertex is at precisely distance $k$ from some dominated vertex. By integrating graph-theoretic techniques, combinatorial optimization, and asymptotic analysis—supported by constructive proofs and counting arguments—the work establishes, for the first time, tight bounds on $\delta_k$ for any fixed $k \geq 3$: specifically, $\frac{1}{4k} \leq \delta_k \leq \frac{k-1}{3k^2 - k - 1}$. Moreover, it exactly determines $\delta_2 = \frac{1}{9}$. These results resolve a longstanding theoretical gap concerning exact distance domination in grid graphs for general $k$.

Abstract

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 lower and upper bounds coincide asymptotically, giving \(δ_2=1/9\).

Problem

• exact-distance domination
• grid graphs
• dominating set
• asymptotic density

Innovation

• exact-distance domination
• grid graphs
• asymptotic density
• domination number
• combinatorial optimization