Cops and Robbers: A $\times$-homotopy Invariant Variant
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866910051995746304 |
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| author | Chih, Tien Scull, Laura |
| author_facet | Chih, Tien Scull, Laura |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13192 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cops and Robbers: A $\times$-homotopy Invariant Variant Chih, Tien Scull, Laura Combinatorics 05C57, 05C60, 05C76, 05C75 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. |
| title | Cops and Robbers: A $\times$-homotopy Invariant Variant |
| topic | Combinatorics 05C57, 05C60, 05C76, 05C75 |
| url | https://arxiv.org/abs/2603.13192 |