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.