Cops and Robber on Hyperbolic Manifolds
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916118476619776 |
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| author | Iršič, Vesna Mohar, Bojan Wesolek, Alexandra |
| author_facet | Iršič, Vesna Mohar, Bojan Wesolek, Alexandra |
| contents | The Cops and Robber game on geodesic spaces is a pursuit-evasion game with discrete steps which captures the behavior of the game played on graphs, as well as that of continuous pursuit-evasion games. One of the outstanding open problems about the game on graphs is to determine which graphs embeddable in a surface of genus $g$ have largest cop number. It is known that the cop number of genus $g$ graphs is $O(g)$ and that there are examples whose cop number is $\tildeΩ(\sqrt{g}\,)$. The same phenomenon occurs when the game is played on geodesic surfaces. In this paper we obtain a surprising result about the game on a surface with constant curvature. It is shown that two cops have a strategy to come arbitrarily close to the robber, independently of the genus. We also discuss upper bounds on the number of cops needed to catch the robber. Our results generalize to higher-dimensional hyperbolic manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05753 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cops and Robber on Hyperbolic Manifolds Iršič, Vesna Mohar, Bojan Wesolek, Alexandra Combinatorics Computational Geometry Metric Geometry 91A05, 91A50, 32Q45 The Cops and Robber game on geodesic spaces is a pursuit-evasion game with discrete steps which captures the behavior of the game played on graphs, as well as that of continuous pursuit-evasion games. One of the outstanding open problems about the game on graphs is to determine which graphs embeddable in a surface of genus $g$ have largest cop number. It is known that the cop number of genus $g$ graphs is $O(g)$ and that there are examples whose cop number is $\tildeΩ(\sqrt{g}\,)$. The same phenomenon occurs when the game is played on geodesic surfaces. In this paper we obtain a surprising result about the game on a surface with constant curvature. It is shown that two cops have a strategy to come arbitrarily close to the robber, independently of the genus. We also discuss upper bounds on the number of cops needed to catch the robber. Our results generalize to higher-dimensional hyperbolic manifolds. |
| title | Cops and Robber on Hyperbolic Manifolds |
| topic | Combinatorics Computational Geometry Metric Geometry 91A05, 91A50, 32Q45 |
| url | https://arxiv.org/abs/2402.05753 |