Cops and Attacking Robbers with Cycle Constraints
<|reference_start|>Cops and Attacking Robbers with Cycle Constraints: 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, Pra\l{}at, and Stacho \cite{bonato2014robber}. We conclude with a list of conjectures and open problems.<|reference_end|>
@article{clow2024cops,
    title={Cops and Attacking Robbers with Cycle Constraints},
    author={Alexander Clow and Melissa A. Huggan and M.E. Messinger},
    journal={arXiv preprint arXiv:2408.02225},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.02225},
    primaryClass={math.CO cs.DM}
}