Cops and Attacking Robbers with Cycle Constraints
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909279761465344 |
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| author | Clow, Alexander Huggan, Melissa A. Messinger, M. E. |
| author_facet | Clow, Alexander Huggan, Melissa A. Messinger, M. E. |
| contents | This paper considers the Cops and Attacking Robbers game, a variant of Cops and Robbers, where the robber is empowered to attack a cop in the same way a cop can capture the robber. In a graph $G$, the number of cops required to capture a robber in the Cops and Attacking Robbers game is denoted by $\attCop(G)$. We characterise the triangle-free graphs $G$ with $\attCop(G) \leq 2$ via a natural generalisation of the cop-win characterisation by Nowakowski and Winkler \cite{nowakowski1983vertex}. We also prove that all bipartite planar graphs $G$ have $\attCop(G) \leq 4$ and show this is tight by constructing a bipartite planar graph $G$ with $\attCop(G) = 4$. Finally we construct $17$ non-isomorphic graphs $H$ of order $58$ with $\attCop(H) = 6$ and $\cop(H)=3$. This provides the first example of a graph $H$ with $\attCop(H) - \cop(H) \geq 3$ extending work by Bonato, Finbow, Gordinowicz, Haidar, Kinnersley, Mitsche, Prałat, and Stacho \cite{bonato2014robber}. We conclude with a list of conjectures and open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02225 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cops and Attacking Robbers with Cycle Constraints Clow, Alexander Huggan, Melissa A. Messinger, M. E. Combinatorics Discrete Mathematics 05C57, 05C75 This paper considers the Cops and Attacking Robbers game, a variant of Cops and Robbers, where the robber is empowered to attack a cop in the same way a cop can capture the robber. In a graph $G$, the number of cops required to capture a robber in the Cops and Attacking Robbers game is denoted by $\attCop(G)$. We characterise the triangle-free graphs $G$ with $\attCop(G) \leq 2$ via a natural generalisation of the cop-win characterisation by Nowakowski and Winkler \cite{nowakowski1983vertex}. We also prove that all bipartite planar graphs $G$ have $\attCop(G) \leq 4$ and show this is tight by constructing a bipartite planar graph $G$ with $\attCop(G) = 4$. Finally we construct $17$ non-isomorphic graphs $H$ of order $58$ with $\attCop(H) = 6$ and $\cop(H)=3$. This provides the first example of a graph $H$ with $\attCop(H) - \cop(H) \geq 3$ extending work by Bonato, Finbow, Gordinowicz, Haidar, Kinnersley, Mitsche, Prałat, and Stacho \cite{bonato2014robber}. We conclude with a list of conjectures and open problems. |
| title | Cops and Attacking Robbers with Cycle Constraints |
| topic | Combinatorics Discrete Mathematics 05C57, 05C75 |
| url | https://arxiv.org/abs/2408.02225 |