Skip to main content
SpringerLink
Log in

Cops and Robbers on Multi-Layer Graphs

  • Conference paper
  • First Online:
Graph-Theoretic Concepts in Computer Science (WG 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14093))

Included in the following conference series:

Abstract

We generalise the popular cops and robbers game to multi-layer graphs, where each cop and the robber are restricted to a single layer (or set of edges). We show that initial intuition about the best way to allocate cops to layers is not always correct, and prove that the multi-layer cop number is neither bounded from above nor below by any function of the cop numbers of the individual layers. We determine that it is NP-hard to decide if k cops are sufficient to catch the robber, even if each layer is a tree plus some isolated vertices. However, we give a polynomial time algorithm to determine if k cops can win when the robber layer is a tree. Additionally, we investigate a question of worst-case division of a simple graph into layers: given a simple graph G, what is the maximum number of cops required to catch a robber over all multi-layer graphs where each edge of G is in at least one layer and all layers are connected? For cliques, suitably dense random graphs, and graphs of bounded treewidth, we determine this parameter up to multiplicative constants. Lastly we consider a multi-layer variant of Meyniel’s conjecture, and show the existence of an infinite family of graphs whose multi-layer cop number is bounded from below by a constant times \(n / \log n\), where n is the number of vertices in the graph.

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 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
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

Notes

  1. 1.

    See [12, Chapter 14] for background on the strong exponential time hypothesis.

References

  1. Aigner, M., Fromme, M.: A game of cops and robbers. Discret. Appl. Math. 8(1), 1–12 (1984)

    Google Scholar 

  2. Alon, N., Spencer, J.H.: The Probabilistic Method, Third Edition. Wiley-Interscience series in discrete mathematics and optimization. Wiley, Hoboken (2008)

    Google Scholar 

  3. Balev, S., Laredo Jiménez, J.L., Lamprou, I., Pigné, Y., Sanlaville, E.: Cops and robbers on dynamic graphs: offline and online case. In: Richa, A.W., Scheideler, C. (eds.) SIROCCO 2020. LNCS, vol. 12156, pp. 203–219. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-54921-3_12

    Chapter  MATH  Google Scholar 

  4. Bollobás, B.: Random graphs, volume 73 of Cambridge Studies in Advanced Mathematics, second edition. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  5. Bonato, A., Nowakowski, R.J.: The Game of Cops and Robbers on Graphs. Student Mathematical Library. American Mathematical Society, New York (2011)

    Google Scholar 

  6. Bonato, A., Pralat, P.: Graph Searching Games and Probabilistic Methods. Discrete Mathematics and Its Applications. CRC Press, London, England, December 2017

    Google Scholar 

  7. Bonato, A., Pralat, P., Wang, C.: Pursuit-evasion in models of complex networks. Internet Math. 4(4), 419–436 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bowler, N.J., Erde, J., Lehner, F., Pitz, M.: Bounding the cop number of a graph by its genus. SIAM J. Discret. Math. 35(4), 2459–2489 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brandt, S., Pettie, S., Uitto, J.: Fine-grained lower bounds on cops and robbers. In: Azar, Y., Bast, H., Herman, G. (eds.), 26th Annual European Symposium on Algorithms (ESA 2018), volume 112 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 9:1–9:12, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2018)

    Google Scholar 

  10. Chiniforooshan, E.: A better bound for the cop number of general graphs. J. Graph Theory 58(1), 45–48 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Clarke, N.E.B.: Constrained cops and robber. ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)-Dalhousie University (Canada) (2002)

    Google Scholar 

  12. Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3

    Book  MATH  Google Scholar 

  13. Enright, J., Meeks, K., Pettersson, W., Sylvester, J.: Cops and robbers on multi-layer graphs. arXiv:2303.03962 (2023)

  14. Fitzpatrick, S.L.: Aspects of domination and dynamic domination. ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)-Dalhousie University (Canada) (1997)

    Google Scholar 

  15. Frankl, P.: Cops and robbers in graphs with large girth and Cayley graphs. Discret. Appl. Math. 17(3), 301–305 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Frieze, A.M., Krivelevich, M., Loh, P.-S.: Variations on cops and robbers. J. Graph Theory 69(4), 383–402 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Joret, G., Kaminski, M., Theis, D.O.: The cops and robber game on graphs with forbidden (induced) subgraphs. Contrib. Discret. Math. 5(2) (2010)

    Google Scholar 

  18. Kinnersley, W.B.: Cops and robbers is exptime-complete. J. Comb. Theory Ser. B 111, 201–220 (2015)

    Google Scholar 

  19. Lehner, F.: On the cop number of toroidal graphs. J. Comb. Theory, Ser. B 151, 250–262 (2021)

    Google Scholar 

  20. Linyuan, L., Peng, X.: On Meyniel’s conjecture of the cop number. J. Graph Theory 71(2), 192–205 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Luczak, T., Pralat, P.: Chasing robbers on random graphs: zigzag theorem. Random Struct. Algorithms 37(4), 516–524 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nowakowski, R.J., Winkler, P.: Vertex-to-vertex pursuit in a graph. Discret. Math. 43(2–3), 235–239 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Petr, J., Portier, J., Versteegen, L.: A faster algorithm for cops and robbers. Discret. Appl. Math. 320, 11–14 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pralat, P., Wormald, N.C.: Meyniel’s conjecture holds for random graphs. Random Struct. Algorithms 48(2), 396–421 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  26. Quilliot, A.: A short note about pursuit games played on a graph with a given genus. J. Comb. Theory Ser. B 38(1), 89–92 (1985)

    Google Scholar 

  27. Schroeder, B.S.W.: The copnumber of a graph is bounded by \(\lfloor \frac{3}{2}\) genus \((G)\rfloor +3\). In: Categorical perspectives (Kent, OH, 1998), Trends Math., pp. 243–263. Birkhäuser Boston, Boston, MA (2001)

    Google Scholar 

  28. Scott, A., Sudakov, B.: A bound for the cops and robbers problem. SIAM J. Discret. Math. 25(3), 1438–1442 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Seymour, P.D., Thomas, R.: Graph searching and a min-max theorem for tree-width. J. Comb. Theory Ser. B 58(1), 22–33 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sipser, M.: Introduction to the Theory of Computation. Cengage Learning, Boston (2012)

    Google Scholar 

  31. Soifer, A.: The Mathematical Coloring Book. Springer, New York (2009). https://doi.org/10.1007/978-0-387-74642-5

    Book  MATH  Google Scholar 

  32. Toruńczyk, S.: Flip-width: cops and robber on dense graphs. In: 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS 2023), page to appear. IEEE (2023)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the Engineering and Physical Sciences Research Council [EP/T004878/1].

Author information

Authors and Affiliations

  1. School of Computing Science, University of Glasgow, Glasgow, UK

    Jessica Enright, Kitty Meeks, William Pettersson & John Sylvester

  2. Department of Computer Science, University of Liverpool, Liverpool, UK

    John Sylvester

Authors
  1. Jessica Enright
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kitty Meeks
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. William Pettersson
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. John Sylvester
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to William Pettersson .

Editor information

Editors and Affiliations

  1. Durham University, Durham, UK

    Daniël Paulusma

  2. University of Fribourg, Fribourg, Fribourg, Switzerland

    Bernard Ries

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

Enright, J., Meeks, K., Pettersson, W., Sylvester, J. (2023). Cops and Robbers on Multi-Layer Graphs. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-43380-1_23

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-43379-5

  • Online ISBN: 978-3-031-43380-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation