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.

在使用图表进行的永恒统治游戏中，攻击者每转一圈都会攻击一个顶点，并且一组守卫必须将一名守卫移到被攻击的顶点处以进行防御。防护装置只能在转弯时移动到相邻的顶点。目的是确定图的永恒控制数\（\ gamma ^ {\ infty} _ {all} \），这是防御无限次攻击序列所需的最少防护数量。本文首先继续研究强网格\（P_n \ boxtimes P_m \）上的永恒统治博弈。笛卡尔网格\（P_n \ square P_m \）已被广泛研究，对于小网格（例如\ {k \ times n \}网格\\（k \ in \ {2,3,4,5 \} \）\ ）。最近证明\（\ gamma ^ {\ infty} _ {all}（P_n \ square P_m）= \ gamma（P_n \ square P_m）+ O（n + m）\）其中\（\ gamma（P_n \ square P_m）\）是\（P_n \ square P_m \）的支配数，它限制了永恒的支配数[Lamprou等。永远统治着大型网格。理论计算机科学，794：27-46，2019]。我们证明，对于\ mathbb {N ^ *} \中的所有\（n，m \）使得\（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}）\）（请注意\（\ lceil \ frac {n} {3} \ rceil \ lceil \ frac { m} {3} \ rceil \）是\（P_n \ boxtimes P_m \））。然后，我们对技术进行概括，以证明\（\ gamma _ {all} ^ {\ infty}（G）= \ gamma（G）+ o（\ gamma（G））\）对于所有图\（G \ in { \ mathcal {F}} \），其中\（{\ mathcal {F}} \）是D维网格的一大族，它们是D维笛卡尔网格的超图和D维强网格的子图。特别地，\（{\ mathcal {F}} \）包括D维笛卡尔网格和D维强网格。