Eternal Domination in Grids

Abstract

In the eternal domination game played on graphs, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices on their turn. The goal is to determine the eternal domination number $\gamma ^{\infty }_{all}$ of a graph which is the minimum number of guards required to defend against an infinite sequence of attacks.

This paper continues the study of the eternal domination game on strong grids $P_n\boxtimes P_m$. Cartesian grids $P_n \,\square \, P_m$ have been vastly studied with tight bounds existing for small grids such as $k\times n$ grids for $k\in \{2,3,4,5\}$. It was recently proven that $\gamma ^{\infty }_{all}(P_n \,\square \, P_m)=\gamma (P_n \,\square \, P_m)+O(n+m)$ where $\gamma (P_n \,\square \, P_m)$ is the domination number of $P_n \,\square \, P_m$ which lower bounds the eternal domination number [Lamprou et al. CIAC 2017]. We prove that, for all $n,m\in \mathbb {N^*}$ such that $m\ge n$, $\lfloor \frac{n}{3}\rfloor \lfloor \frac{m}{3}\rfloor +{\varOmega }(n+m)=\gamma _{all}^{\infty } (P_{n}\boxtimes P_{m})=\lceil \frac{n}{3}\rceil \lceil \frac{m}{3}\rceil + O(m\sqrt{n})$ (note that $\lceil \frac{n}{3}\rceil \lceil \frac{m}{3}\rceil$ is the domination number of $P_n\boxtimes P_m$). Our technique may be applied to other “grid-like” graphs.

This work has been partially supported by ANR program “Investments for the Future” under reference ANR-11-LABX-0031-01, the Inria Associated Team AlDyNet. Due to a lack of space, several proofs have been omitted and can be found in [14].