Skip to main content
SpringerLink
Log in

Cops and Robbers on 1-Planar Graphs

  • Conference paper
  • First Online:
Graph Drawing and Network Visualization (GD 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14466))

Included in the following conference series:

  • 99 Accesses

Abstract

Cops and Robbers is a well-studied pursuit-evasion game in which a set of cops seeks to catch a robber in a graph G, where cops and robber move along edges of G. The cop number of G is the minimum number of cops that is sufficient to catch the robber. Every planar graph has cop number at most three, and there are planar graphs for which three cops are necessary [Aigner and Fromme, DAM 1984]. We study the problem for 1-planar graphs, that is, graphs that can be drawn in the plane with at most one crossing per edge. In contrast to planar graphs, we show that some 1-planar graphs have unbounded cop number. Meanwhile, for maximal 1-planar graphs, we prove that three cops are always sufficient and sometimes necessary. In addition, we completely determine the cop number of outer 1-planar graphs.

This research is funded in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aigner, M., Fromme, M.: A game of cops and robbers. Discrete Applied Mathematics 8(1), 1–12 (1984)

    Article  MathSciNet  Google Scholar 

  2. Auer, C., Bachmaier, C., Brandenburg, F.J., Gleißner, A., Hanauer, K., Neuwirth, D., Reislhuber, J.: Outer \(1\)-planar graphs. Algorithmica 74(4), 1293–1320 (2016)

    Article  MathSciNet  Google Scholar 

  3. Barát, J., Tóth, G.: Improvements on the density of maximal 1-planar graphs. Journal of Graph Theory 88(1), 101–109 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bekos, M.A., Didimo, W., Liotta, G., Mehrabi, S., Montecchiani, F.: On rac drawings of 1-planar graphs. Theoret. Comput. Sci. 689, 48–57 (2017) 10.1016/j.tcs.2017.05.039, https://www.sciencedirect.com/science/article/pii/S0304397517304851

  5. Berarducci, A., Intrigila, B.: On the cop number of a graph. Advances in Applied Mathematics 14, 389–403 (1993)

    Article  MathSciNet  Google Scholar 

  6. Bonato, A., Nowakowski, R.: The Game of Cops and Robbers on Graphs. American Mathematical Society, Student mathematical library (2011)

    Book  Google Scholar 

  7. Bonato, A., Breen, J., Brimkov, B., Carlson, J., English, S., Geneson, J.T., Hogben, L., Perry, K.E., Reinhart, C.: Optimizing the trade-off between number of cops and capture time in cops and robbers. Journal of Combinatorics 13(1), 79–203 (2019)

    Article  MathSciNet  Google Scholar 

  8. Bonato, A., Mohar, B.: Topological directions in cops and robbers. Journal of Combinatorics 11, 47–64 (2020)

    Article  MathSciNet  Google Scholar 

  9. Brandenburg, F.J., Eppstein, D., Gleißner, A., Goodrich, M.T., Hanauer, K., Reislhuber, J.: On the density of maximal 1-planar graphs. In: Graph Drawing. pp. 327–338. Springer, Berlin Heidelberg, Berlin, Heidelberg (2013)

    Google Scholar 

  10. Chaplick, S., Kryven, M., Liotta, G., Löffler, A., Wolff, A.: Beyond outerplanarity. In: In Proceedings of the International Symposium on Graph Drawing and Network Visualization (GD). pp. 546–559. Springer (2018)

    Google Scholar 

  11. Clarke, N.E., MacGillivray, G.: Characterizations of k-cop win graphs. Discrete Mathematics 312(8), 1421–1425 (2012)

    Article  MathSciNet  Google Scholar 

  12. Didimo, W., Liotta, G., Montecchiani, F.: A survey on graph drawing beyond planarity. ACM Comput. Surv. 52(1), 1–37 (2019)

    Article  Google Scholar 

  13. Durocher, S., Kamali, S., Kryven, M., Liu, F., Mashghdoust, A., Miller, A., Nezhad, P.Z., Costa, I.P., Zapp, T.: Cops and robbers on 1-planar graphs (2023), arXiv:2309.01001

  14. Eppstein, D.: \(k\)-planar graphs and genus. Computer Science Stack Exchange (2015), https://mathoverflow.net/questions/221968/k-planar-graphs-and-genus Accessed: 2023–06-06

  15. Erde, J., Lehner, F.: Improved bounds on the cop number of a graph drawn on a surface. In: Extended Abstracts of the European Conference on Combinatorics, Graph Theory and Applications (EuroComb). pp. 111–116. Springer International Publishing (2021)

    Google Scholar 

  16. Fomin, F.V., Golovach, P.A., Kratochvíl, J., Nisse, N., Suchan, K.: Pursuing a fast robber on a graph. Theoret. Comput. Sci. 411(7), 1167–1181 (2010) 10.1016/j.tcs.2009.12.010, https://www.sciencedirect.com/science/article/pii/S0304397509008342

  17. Gahlawat, H., Zehavi, M.: Parameterized Analysis of the Cops and Robber Game. In: Leroux, J., Lombardy, S., Peleg, D. (eds.) 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), vol. 272, pp. 49:1–49:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2023). https://doi.org/10.4230/LIPIcs.MFCS.2023.49, https://drops.dagstuhl.de/opus/volltexte/2023/18583

  18. Grigoriev, A., Bodlaender, H.L.: Algorithms for graphs embeddable with few crossings per edge. Algorithmica 49(1), 1–11 (2007)

    Article  MathSciNet  Google Scholar 

  19. Hong, S.H., Tokuyama, T.: Beyond Planar Graphs. Communications of NII Shonan Meetings (2020)

    Google Scholar 

  20. Joret, G., Kamiński, M., Theis, D.O.: The cops and robber game on graphs with forbidden (induced) subgraphs. Contributions to Discrete Mathematics 5(2), 40–51 (2010)

    MathSciNet  Google Scholar 

  21. Kalai, G., Nevo, E., Novik, I.: Bipartite rigidity. Transactions of the American Mathematical Society 368(8), 5515–5545 (2016), https://www.jstor.org/stable/tranamermathsoci.368.8.5515

  22. Korzhik, V.P., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. In: Proceedings of the 16th International Symposium on Graph Drawing (GD). Lecture Notes in Computer Science, vol. 5417, pp. 302–312. Springer (2008)

    Google Scholar 

  23. Lubiw, A., Snoeyink, J., Vosoughpour, H.: Visibility graphs, dismantlability, and the cops and robbers game. Computational Geometry 66, 14–27 (2017)

    Article  MathSciNet  Google Scholar 

  24. Maurer, A., McCauley, J., Valeva, S., Beveridge, A., Islar, V., Saraswat, V.: Cops and Robbers on Planar Graphs. In: Summer 2010 Interdisciplinary Research Experience for Undergraduates (2010)

    Google Scholar 

  25. Neufeld, S., Nowakowski, R.: A game of cops and robbers played on products of graphs. Discrete Mathematics 186(1), 253–268 (1998)

    Article  MathSciNet  Google Scholar 

  26. Nowakowski, R.J., Winkler, P.: Vertex-to-vertex pursuit in a graph. Discrete Mathematics 43, 235–239 (1983)

    Article  MathSciNet  Google Scholar 

  27. Quilliot, A.: Jeux et pointes fixes sur les graphes. Ph.D. thesis, Université de Paris VI (1978)

    Google Scholar 

  28. Quilliot, A.: A short note about pursuit games played on a graph with a given genus. Journal of Combinatorial Theory, Series B 38(1), 89–92 (1985)

    Article  MathSciNet  Google Scholar 

  29. Ringel, G.: Ein sechsfarbenproblem auf der kugel. In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. vol. 29, pp. 107–117. Springer (1965)

    Google Scholar 

  30. Schröder, B.S.W.: The Copnumber of a Graph is Bounded by [3/2 genus (G)] + 3, pp. 243–263. Birkhäuser Boston (2001)

    Google Scholar 

  31. Schumacher, V.H.: Zur struktur 1-planarer graphen. Mathematische Nachrichten 125(1), 291–300 (1986) 10.1002/mana.19861250122, https://onlinelibrary.wiley.com/doi/abs/10.1002/mana.19861250122

  32. Schumacher, V.H.: Zur struktur 1-planarer graphen. Mathematische Nachrichten 125(1), 291–300 (1986)

    Article  MathSciNet  Google Scholar 

  33. Suzuki, Y.: Re-embeddings of maximum 1-planar graphs. SIAM Journal on Discrete Mathematics 24, 1527–1540 (2010)

    Article  MathSciNet  Google Scholar 

  34. Wood, D.R., Telle, J.A.: Planar decompositions and the crossing number of graphs with an excluded minor. In: Proceedings of the 14th International Symposium on Graph Drawing (GD). pp. 150–161. Springer (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department Computer Science, University of Manitoba, Winnipeg, MB, Canada

    Stephane Durocher, Myroslav Kryven, Fengyi Liu, Amirhossein Mashghdoust, Avery Miller, Pouria Zamani Nezhad, Ikaro Penha Costa & Timothy Zapp

  2. Department Electrical Engineering and Computer Science, York University, Toronto, ON, Canada

    Shahin Kamali

Authors
  1. Stephane Durocher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shahin Kamali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Myroslav Kryven
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fengyi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Amirhossein Mashghdoust
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Avery Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Pouria Zamani Nezhad
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Ikaro Penha Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Timothy Zapp
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fengyi Liu .

Editor information

Editors and Affiliations

  1. University of Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. Osnabrück University, Osnabrück, Germany

    Markus Chimani

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Durocher, S. et al. (2023). Cops and Robbers on 1-Planar Graphs. In: Bekos, M.A., Chimani, M. (eds) Graph Drawing and Network Visualization. GD 2023. Lecture Notes in Computer Science, vol 14466. Springer, Cham. https://doi.org/10.1007/978-3-031-49275-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-49275-4_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-49274-7

  • Online ISBN: 978-3-031-49275-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation