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Cops and Robbers for Graphs on Surfaces with Crossings

Authors Prosenjit Bose , Pat Morin , Karthik Murali

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ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • Cops and Robbers
  • Crossings
  • 1-Planar
  • Surfaces

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Abstract

Cops and Robbers is a game played on a graph where a set of cops attempt to capture a single robber. The game proceeds in rounds, where each round first consists of the cops' turn, followed by the robber’s turn. In the first round, the cops place themselves on a subset of vertices, after which the robber chooses a vertex to place himself. From the next round onwards, in the cops' turn, every cop can choose to either stay on the same vertex or move to an adjacent vertex, and likewise the robber in his turn. The robber is considered to be captured if, at any point in time, there is some cop on the same vertex as the robber. The cops win if they can capture the robber within a finite number of rounds; else the robber wins. 
A natural question in this game concerns the cop-number of a graph - the minimum number of cops needed to capture a robber. It has long been known that graphs embeddable (without crossings) on surfaces of bounded genus have bounded cop-number. In contrast, it was shown recently that the class of 1-planar graphs - graphs that can be drawn on the plane with at most one crossing per edge - does not have bounded cop-number. This paper initiates an investigation into how the distance between crossing pairs of edges influences a graph’s cop number. In particular, we look at Distance d Cops and Robbers, a variant of the classical game, where the robber is considered to be captured if there is a cop within distance d of the robber.
Let c_d(G) denote the minimum number of cops required in the graph G to capture a robber within distance d. We look at various classes of graphs, such as 1-plane graphs, k-plane graphs (graphs where each edge is crossed at most k times), and even general graph drawings, and show that if every crossing pair of edges can be connected by a path of small length, then c_d(G) is bounded, for small values of d. For example, we show that if a graph G admits a drawing in which every pair of crossing edges is contained in a path of length at most 3, then c₄(G) ≤ 21. And if the drawing permits a stronger assumption that the endpoints of every crossing induce the complete graph K₄, then c₃(G) ≤ 9. The tools and techniques that we develop in this paper are sufficiently general, enabling us to examine graphs drawn not only on the sphere but also on orientable and non-orientable surfaces.

Cite As Get BibTex

Prosenjit Bose, Pat Morin, and Karthik Murali. Cops and Robbers for Graphs on Surfaces with Crossings. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.27

Author Details

Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Pat Morin
  • School of Computer Science, Carleton University, Ottawa, Canada
Karthik Murali
  • School of Computer Science, Carleton University, Ottawa, Canada

Funding

Research supported in part by NSERC.

References

  1. M. Aigner and M. Fromme. A game of cops and robbers. Discrete Applied Mathematics, 8(1):1-12, 1984. URL: https://doi.org/10.1016/0166-218X(84)90073-8.
  2. Thomas Andreae. Note on a pursuit game played on graphs. Discret. Appl. Math., 9(2):111-115, 1984. URL: https://doi.org/10.1016/0166-218X(84)90012-X.
  3. Thomas Andreae. On a pursuit game played on graphs for which a minor is excluded. J. Comb. Theory, Ser. B, 41(1):37-47, 1986. URL: https://doi.org/10.1016/0095-8956(86)90026-2.
  4. William Baird and Anthony Bonato. Meyniel’s conjecture on the cop number: A survey. Journal of Combinatorics, 3(2):225-238, 2012. URL: https://doi.org/10.4310/joc.2012.v3.n2.a6.
  5. Michael A. Bekos, Michael Kaufmann, and Chrysanthi N. Raftopoulou. On optimal 2- and 3-planar graphs. In Boris Aronov and Matthew J. Katz, editors, 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, volume 77 of LIPIcs, pages 16:1-16:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPICS.SOCG.2017.16.
  6. Michael A. Bekos, Giordano Da Lozzo, Svenja M. Griesbach, Martin Gronemann, Fabrizio Montecchiani, and Chrysanthi N. Raftopoulou. Book embeddings of k-framed graphs and k-map graphs. Discret. Math., 347(1):113690, 2024. URL: https://doi.org/10.1016/j.disc.2023.113690.
  7. Alessandro Berarducci and Benedetto Intrigila. On the cop number of a graph. Advances in Applied mathematics, 14(4):389-403, 1993. Google Scholar
  8. Andrew Beveridge, Andrzej Dudek, Alan M. Frieze, and Tobias Müller. Cops and robbers on geometric graphs. Comb. Probab. Comput., 21(6):816-834, 2012. URL: https://doi.org/10.1017/S0963548312000338.
  9. Sourabh Bhattacharya, Salvatore Candido, and Seth Hutchinson. Motion Strategies for Surveillance, pages 249-256. The MIT Press, January 2008. URL: https://doi.org/10.7551/mitpress/7830.003.0033.
  10. Therese Biedl, Prosenjit Bose, and Karthik Murali. A parameterized algorithm for vertex and edge connectivity of embedded graphs. In 32nd Annual European Symposium on Algorithms, ESA 2024, September 2-4, 2024, Royal Holloway, London, United Kingdom, volume 308 of LIPIcs, pages 24:1-24:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ESA.2024.24.
  11. Therese Biedl and Karthik Murali. On computing the vertex connectivity of 1-plane graphs. In 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 23:1-23:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.23.
  12. Anthony Bonato, Ehsan Chiniforooshan, and Pawel Pralat. Cops and robbers from a distance. Theor. Comput. Sci., 411(43):3834-3844, 2010. URL: https://doi.org/10.1016/J.TCS.2010.07.003.
  13. Anthony Bonato and Bojan Mohar. Topological directions in cops and robbers. Journal of Combinatorics, 11(1):47-64, 2020. URL: https://doi.org/10.4310/joc.2020.v11.n1.a3.
  14. Anthony Bonato and Richard Nowakowski. The Game of Cops and Robbers on Graphs. American Mathematical Society, August 2011. URL: https://doi.org/10.1090/stml/061.
  15. Anthony Bonato, Pawel Pralat, and Changping Wang. Pursuit-evasion in models of complex networks. Internet Math., 4(4):419-436, 2007. URL: https://doi.org/10.1080/15427951.2007.10129149.
  16. Prosenjit Bose, Jean-Lou De Carufel, Anil Maheshwari, and Karthik Murali. On 1-planar graphs with bounded cop-number. Theor. Comput. Sci., 1037:115160, 2025. URL: https://doi.org/10.1016/J.TCS.2025.115160.
  17. Prosenjit Bose, Pat Morin, and Karthik Murali. Cops and robbers for graphs on surfaces with crossings. CoRR, abs/2504.13813, 2025. URL: https://doi.org/10.48550/arXiv.2504.13813.
  18. Nathan J. Bowler, Joshua Erde, Florian Lehner, and Max Pitz. Bounding the cop number of a graph by its genus. SIAM J. Discret. Math., 35(4):2459-2489, 2021. URL: https://doi.org/10.1137/20M1312150.
  19. Franz J. Brandenburg. Characterizing and recognizing 4-map graphs. Algorithmica, 81(5):1818-1843, 2019. URL: https://doi.org/10.1007/S00453-018-0510-X.
  20. Zhi-Zhong Chen, Michelangelo Grigni, and Christos H. Papadimitriou. Map graphs. J. ACM, 49(2):127-138, 2002. URL: https://doi.org/10.1145/506147.506148.
  21. Timothy H. Chung, Geoffrey A. Hollinger, and Volkan Isler. Search and pursuit-evasion in mobile robotics - A survey. Auton. Robots, 31(4):299-316, 2011. URL: https://doi.org/10.1007/S10514-011-9241-4.
  22. Nancy E. Clarke, Samuel Fiorini, Gwenaël Joret, and Dirk Oliver Theis. A note on the cops and robber game on graphs embedded in non-orientable surfaces. Graphs Comb., 30(1):119-124, 2014. URL: https://doi.org/10.1007/S00373-012-1246-Z.
  23. Stephane Durocher, Shahin Kamali, Myroslav Kryven, Fengyi Liu, Amirhossein Mashghdoust, Avery Miller, Pouria Zamani Nezhad, Ikaro Penha Costa, and Timothy Zapp. Cops and robbers on 1-planar graphs. In Graph Drawing and Network Visualization - 31st International Symposium, GD 2023, Isola delle Femmine, Palermo, Italy, September 20-22, 2023, Revised Selected Papers, Part II, volume 14466 of Lecture Notes in Computer Science, pages 3-17. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-49275-4_1.
  24. Tomas Gavenciak, Przemyslaw Gordinowicz, Vít Jelínek, Pavel Klavík, and Jan Kratochvíl. Cops and robbers on intersection graphs. Eur. J. Comb., 72:45-69, 2018. URL: https://doi.org/10.1016/J.EJC.2018.04.009.
  25. Jonathan L Gross and Thomas W Tucker. Topological graph theory. Courier Corporation, 2001. Google Scholar
  26. Gwenaël Joret, Marcin Kaminski, and Dirk Oliver Theis. The cops and robber game on graphs with forbidden (induced) subgraphs. Contributions Discret. Math., 5(2), 2010. URL: https://doi.org/10.11575/CDM.V5I2.62032.
  27. Richard J. Nowakowski and Peter Winkler. Vertex-to-vertex pursuit in a graph. Discret. Math., 43(2-3):235-239, 1983. URL: https://doi.org/10.1016/0012-365X(83)90160-7.
  28. RJ Nowakowski and BSW Schröder. Bounding the cop number using the crosscap number. Unpublished note, 1997. Google Scholar
  29. Alain Quilliot. Jeux et pointes fixes sur les graphes. PhD thesis, Université de Paris VI, 1978. Google Scholar
  30. Alain Quilliot. A short note about pursuit games played on a graph with a given genus. J. Comb. Theory, Ser. B, 38(1):89-92, 1985. URL: https://doi.org/10.1016/0095-8956(85)90093-0.
  31. Bernd SW Schröder. The copnumber of a graph is bounded by ⌊3/2 genus(G)⌋ + 3. Categorical perspectives, pages 243-263, 2001. URL: https://doi.org/10.1007/978-1-4612-1370-3_14.
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