One-visibility cops and robber on trees

Yang, Boting; Akter, Tanzina

doi:10.1016/j.tcs.2021.07.038

Cited by 4 publications

(2 citation statements)

References 19 publications

(21 reference statements)

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“…Among the many variants of the game of Cops and Robbers, one theme is to limit the visibility of the robber. For a nonnegative integer k, in k-visibility Cops and Robbers, the robber is visible to the cops only when a cop is distance at most k. The case when k = 0 has been studied [17,18,23], as has the case when k = 1 [26,27,28], and a recent paper covers the cases k ≥ 1 [16].…”

Section: Introductionmentioning

confidence: 99%

The one-visibility Localization game

Bonato¹,

Marbach²,

Molnar³

et al. 2023

Preprint

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We introduce a variant of the Localization game in which the cops only have visibility one, along with the corresponding optimization parameter, the onevisibility localization number ζ1. By developing lower bounds using isoperimetric inequalities, we give upper and lower bounds for ζ1 on k-ary trees with k ≥ 2 that differ by a multiplicative constant, showing that the parameter is unbounded on kary trees. We provide a O( √ 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 ζ1 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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Section: Introductionmentioning

confidence: 99%

The one-visibility Localization game

Bonato¹,

Marbach²,

Molnar³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Among the many variants of the game of Cops and Robbers, one theme is to limit the visibility of the robber. For a nonnegative integer k, in k-visibility Cops and Robbers, the robber is visible to the cops only when a cop is distance at most k. The case when k = 0 has been studied [16,17,23], as has the case when k = 1 [28,26,27], and a recent paper studied the cases k ≥ 1 [15].…”

Section: Introductionmentioning

confidence: 99%