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arXiv · 2603.13192

Cops and Robbers: A $\times$-homotopy Invariant Variant

Tien Chih·Laura Scull
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Abstract

Cops and Robbers is a pursuit-evasion game played on graphs, of which many variants have been developed and studied. We introduce a variant of this game, "Sneaky-Active Cops and Robbers", where all cops and robber must move on their turn, and where the robber is allowed to move onto a cop position without being captured. We show that for reflexive graphs, this game is equivalent to the classical cops and robbers and that the cop number for a graph is invariant under $\times$-homotopy equivalence. We then develop further properties of this game, computing cop numbers for a number of graph families and developing results about the behavior of categorical and box products of graphs.

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Tien Chih, Laura Scull. 2026-03-13. Cops and Robbers: A $\times$-homotopy Invariant Variant. https://arxiv.org/abs/2603.13192

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