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arXiv · 2509.18516

Cops and robbers on chess graphs

Sally Ambrose·Evan Angelone·Jacob Chen·Daniel Ma·Arturo Ortiz San Miguel·Wraven Watanabe·Stephen Whitcomb·Shanghao Wu
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Abstract

Cops and robbers is a pursuit-evasion game played on graphs. We completely classify the cop numbers for $n \times n$ knight graphs and queen graphs. This completes the classification of the cop numbers for all $n \times n$ classical chess graphs. As a corollary, we resolve an open problem about the monotonicity of $c$($\mathcal{Q}_n$). Moreover, we introduce \emph{royal graphs}, a generalization of chess graphs for arbitrary piece movements, which models real-life movement constraints. We give results on the cop numbers for these families.

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Sally Ambrose, Evan Angelone, Jacob Chen, Daniel Ma, Arturo Ortiz San Miguel, Wraven Watanabe, Stephen Whitcomb, Shanghao Wu. 2025-09-23. Cops and robbers on chess graphs. https://arxiv.org/abs/2509.18516

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