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Reports (Research Report) Year : 2018

## Eternal Domination in Grids

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Fionn Mc Inerney
Nicolas Nisse
Stéphane Pérennes
• Function : Author
• PersonId : 942945

#### 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 [Lamprou et al., CIAC 2017]. We prove that, for all $n,m\in \mathbb{N^*}$ such that $m\geq n$, $\lfloor \frac{n}{3} \rfloor \lfloor \frac{m}{3} \rfloor+\Omega(n+m)=\gamma_{all}^{\infty} (P_{n}\boxtimes P_{m})=\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil + O(m\sqrt{n})$ (note that $\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil$ is the domination number of $P_n\boxtimes P_m$). Our technique may be applied to other grid-like" graphs.

### Dates and versions

hal-01790322 , version 1 (11-05-2018)
hal-01790322 , version 2 (25-07-2018)
hal-01790322 , version 3 (12-04-2019)

### Identifiers

• HAL Id : hal-01790322 , version 3

### Cite

Fionn Mc Inerney, Nicolas Nisse, Stéphane Pérennes. Eternal Domination in Grids. [Research Report] Inria & Université Cote d'Azur, CNRS, I3S, Sophia Antipolis, France. 2018. ⟨hal-01790322v3⟩

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