Perpetually Dominating Large Grids

Lamprou, Ioannis ORCID: 0000-0001-5337-7336, Martin, Russell ORCID: 0000-0002-7043-503X and Schewe, Sven ORCID: 0000-0002-9093-9518 (2017) Perpetually Dominating Large Grids. ALGORITHMS AND COMPLEXITY (CIAC 2017), 10236. pp. 393-404.

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Abstract

In the m-\emph{Eternal Domination} game, a team of guard tokens initially occupies a dominating set on a graph $G$. An attacker then picks a vertex without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked vertex, while each remaining one can choose to move to one of his neighboring vertices. The new guards' placement must again be dominating. This attack-defend procedure continues eternally. The guards win if they can eternally maintain a dominating set against any sequence of attacks, otherwise, the attacker wins. The m-\emph{eternal domination number} for a graph $G$ is the minimum amount of guards such that they win against any attacker strategy in $G$ (all guards move model). We study rectangular grids and provide the first known general upper bound on the m-eternal domination number for these graphs. Our novel strategy implements a square rotation principle and eternally dominates $m \times n$ grids by using approximately $\frac{mn}{5}$ guards, which is asymptotically optimal even for ordinary domination.

Item Type:	Article
Additional Information:	latest full draft version
Uncontrolled Keywords:	Eternal domination, Combinatorial game, Two players, Graph protection, Grid
Depositing User:	Symplectic Admin
Date Deposited:	02 Mar 2017 10:23
Last Modified:	19 Jan 2023 07:15
DOI:	10.1007/978-3-319-57586-5_33
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3006036