King's graph

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.