Eternal Domination: D-Dimensional Cartesian and Strong Grids and Everything in Between

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 first 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. Eternally dominating large grids. Theoretical Computer Science, 794:27–46, 2019]. We prove that, for all $n,m\in \mathbb {N^*}$ such that $m\ge n$, $\lfloor \frac{n}{3} \rfloor \lfloor \frac{m}{3} \rfloor +\Omega (n+m)=\gamma _{all}^{\infty } (P_{n}\boxtimes P_{m})=\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil + O(m\sqrt{n})$ (note that $\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil$ is the domination number of $P_n\boxtimes P_m$). We then generalise our technique to prove that $\gamma _{all}^{\infty }(G)=\gamma (G)+o(\gamma (G))$ for all graphs $G\in {\mathcal {F}}$, where ${\mathcal {F}}$ is a large family of D-dimensional grids which are supergraphs of the D-dimensional Cartesian grid and subgraphs of the D-dimensional strong grid. In particular, ${\mathcal {F}}$ includes both the D-dimensional Cartesian grid and the D-dimensional strong grid.