Eternal Domination in Grids

Fionn Mc Inerney 1 Nicolas Nisse 1 Stéphane Pérennes 1
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
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. We prove that, for all $n,m\in \mathbb{N^*}$ such that $m\geq n$, $\lceil \frac{nm}{9} \rceil+\Omega(n+m)=\gamma_{all}^{\infty} (P_{n}\boxtimes P_{m})=\lceil\frac{nm}{9}\rceil + O(m\sqrt{n})$ (note that $\lceil \frac{nm}{9} \rceil$ is the domination number of $P_n\boxtimes P_m$).
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Fionn Mc Inerney, Nicolas Nisse, Stéphane Pérennes. Eternal Domination in Grids. [Research Report] Inria & Université Nice Sophia Antipolis, CNRS, I3S, Sophia Antipolis, France. 2018. ⟨hal-01790322v2⟩

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