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

  • Klaus-Tycho Förster
  • Rijad Nuridini
  • Jara Uitto
  • Roger Wattenhofer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9439)

Abstract

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 \(\Omega \left(n^2 \cdot \log\left(\ell/n\right) \right)\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aigner, M., Fromme, M.: A Game of Cops and Robbers. Discrete Applied Mathematics 8(1), 1–12 (1984)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alspach, B.: Sweeping and Searching in Graphs: a Brief Survey. Matematiche 59, 5–37 (2006)MathSciNetMATHGoogle Scholar
  3. 3.
    Berarducci, A., Intrigila, B.: On the Cop Number of a Graph. Advances in Applied Mathematics 14(4), 389–403 (1993)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gavenciak, T.: Cop-win Graphs with Maximum Capture-time. Discrete Mathematics 310(10–11), 1557–1563 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hahn, G.: Cops, Robbers and Graphs. Tatra Mt. Math. Publ. 36(163), 163–176 (2007)MathSciNetMATHGoogle Scholar
  16. 16.
    Kinnersley, W.B.: Cops and Robbers is EXPTIME-complete. J. Comb. Theory, Ser. B 111, 201–220 (2015)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mehrabian, A.: The Capture Time of Grids. Discrete Mathematics 311(1), 102–105 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Moon, J.W.: On the Diameter of a Graph. Michigan Math. J. 12(3), 349–351 (1965)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nowakowski, R.J., Winkler, P.: Vertex-to-vertex Pursuit in a Graph. Discrete Mathematics 43(2-3), 235–239 (1983)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Klaus-Tycho Förster
    • 1
  • Rijad Nuridini
    • 1
  • Jara Uitto
    • 1
  • Roger Wattenhofer
    • 1
  1. 1.Computer Engineering and Networks LaboratoryETH ZurichZurichSwitzerland

Personalised recommendations