Lower Bounds for the Capture Time: Linear, Quadratic, and Beyond

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9439)


In the classical game of Cops and Robbers on graphs, the capture time is defined by the least number of moves needed to catch all robbers with the smallest amount of cops that suffice. While the case of one cop and one robber is well understood, it is an open question how long it takes for multiple cops to catch multiple robbers. We show that capturing \(\ell \in {\mathcal{O}}\left(n\right)\) robbers can take \(\Omega\left(\ell \cdot n\right)\) time, inducing a capture time of up to \(\Omega\left(n^2\right)\). For the case of one cop, our results are asymptotically optimal. Furthermore, we consider the case of a superlinear amount of robbers, where we show a capture time of \(\Omega \left(n^2 \cdot \log\left(\ell/n\right) \right)\).


Cayley Graph Graph Construction Capture Time Classical Game Search Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aigner, M., Fromme, M.: A Game of Cops and Robbers. Discrete Applied Mathematics 8(1), 1–12 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alspach, B.: Sweeping and Searching in Graphs: a Brief Survey. Matematiche 59, 5–37 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Berarducci, A., Intrigila, B.: On the Cop Number of a Graph. Advances in Applied Mathematics 14(4), 389–403 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonato, A., Chiniforooshan, E.: Pursuit and evasion from a distance: algorithms and bounds. In: Proceedings of the Sixth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 1–10. SIAM (2009)Google Scholar
  5. 5.
    Bonato, A., Golovach, P.A., Hahn, G., Kratochvíl, J.: The Capture Time of a Graph. Discrete Mathematics 309(18), 5588–5595 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonato, A., Gordinowicz, P., Kinnersley, B., Prałat, P.: The Capture Time of the Hypercube. Electr. J. Comb. 20(2), P24 (2013)Google Scholar
  7. 7.
    Bonato, A., Nowakowski, R.J.: The Game of Cops and Robbers on Graphs. Student Mathematical Library, vol. 61. American Mathematical Society, Providence (2011)zbMATHGoogle Scholar
  8. 8.
    Bonato, A., Yang, B.: Graph searching and related problems. In: Handbook of Combinatorial Optimization, pp. 1511–1558. Springer, New York (2013)CrossRefGoogle Scholar
  9. 9.
    Breisch, R.: An Intuitive Approach to Speleotopology. Southwestern Cavers 6(5), 72–78 (1967)Google Scholar
  10. 10.
    Clarke, N.E., MacGillivray, G.: Characterizations of k-copwin Graphs. Discrete Mathematics 312(8), 1421–1425 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Deo, N., Nikoloski, Z.: The Game of Cops and Robbers on Graphs: a Model for Quarantining Cyber Attacks. Congressus Numerantium, 193–216 (2003)Google Scholar
  12. 12.
    Frankl, P.: Cops and Robbers in Graphs with Large Girth and Cayley Graphs. Discrete Appl. Math. 17(3), 301–305 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Frieze, A.M., Krivelevich, M., Loh, P.-S.: Variations on Cops and Robbers. Journal of Graph Theory 69(4), 383–402 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gavenciak, T.: Cop-win Graphs with Maximum Capture-time. Discrete Mathematics 310(10–11), 1557–1563 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hahn, G.: Cops, Robbers and Graphs. Tatra Mt. Math. Publ. 36(163), 163–176 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kinnersley, W.B.: Cops and Robbers is EXPTIME-complete. J. Comb. Theory, Ser. B 111, 201–220 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kosowski, A., Li, B., Nisse, N., Suchan, K.: k-Chordal graphs: from cops and robber to compact routing via treewidth. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 610–622. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  18. 18.
    Lu, L., Peng, X.: On Meyniel’s Conjecture of the Cop Number. Journal of Graph Theory 71(2), 192–205 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mehrabian, A.: The Capture Time of Grids. Discrete Mathematics 311(1), 102–105 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Moon, J.W.: On the Diameter of a Graph. Michigan Math. J. 12(3), 349–351 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nowakowski, R.J., Winkler, P.: Vertex-to-vertex Pursuit in a Graph. Discrete Mathematics 43(2-3), 235–239 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Parsons, T.D.: Pursuit-evasion in a graph. In: Alavi, Y., Lick, D.R. (eds.) AII 1992. LNCS, vol. 642, pp. 426–441. Springer, Heidelberg (1992)Google Scholar
  23. 23.
    Parsons, T.D.: The search number of a connected graph. In: Proc. 9th Southeast. Conf. on Combinatorics, Graph Theory, and Computing (1978)Google Scholar
  24. 24.
    Prałat, P.: When Does a Random Graph Have a Constant Cop Number. Australasian Journal of Combinatorics 46, 285–296 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Quilliot, A.: Jeux et Pointes Fixes sur les Graphes. Ph.D. thesis, Universite de Paris VI (1978)Google Scholar
  26. 26.
    Scott, A., Sudakov, B.: A Bound for the Cops and Robbers Problem. SIAM J. Discrete Math. 25(3), 1438–1442 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Computer Engineering and Networks LaboratoryETH ZurichZurichSwitzerland

Personalised recommendations