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arXiv · 2301.03534

The one-visibility Localization game

Anthony Bonato·Trent G. Marbach·Michael Molnar·JD Nir
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Abstract

We introduce a variant of the Localization game in which the cops only have visibility one, along with the corresponding optimization parameter, the one-visibility localization number $ζ_1$. By developing lower bounds using isoperimetric inequalities, we give upper and lower bounds for $ζ_1$ on $k$-ary trees with $k\ge 2$ that differ by a multiplicative constant, showing that the parameter is unbounded on $k$-ary trees. We provide a $O(\sqrt{n})$ bound for $K_h$-minor free graphs of order $n$, and we show Cartesian grids meet this bound by determining their one-visibility localization number up to four values. We present upper bounds on $ζ_1$ using pathwidth and the domination number and give upper bounds on trees via their depth and order. We conclude with open problems.

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Anthony Bonato, Trent G. Marbach, Michael Molnar, JD Nir. 2023-01-09. The one-visibility Localization game. https://doi.org/10.1016/j.tcs.2023.114186

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