The Cop Number of the One-Cop-Moves Game on Planar Graphs

  • Ziyuan Gao
  • Boting YangEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10628)


Cops and robbers is a vertex-pursuit game played on graphs. In the classical cops-and-robbers game, a set of cops and a robber occupy the vertices of the graph and move alternately along the graph’s edges with perfect information about each other’s positions. If a cop eventually occupies the same vertex as the robber, then the cops win; the robber wins if she can indefinitely evade capture. Aigner and Frommer established that in every connected planar graph, three cops are sufficient to capture a single robber. In this paper, we consider a recently studied variant of the cops-and-robbers game, alternately called the one-active-cop game, one-cop-moves game or the lazy-cops-and-robbers game, where at most one cop can move during any round. We show that Aigner and Frommer’s result does not generalise to this game variant by constructing a connected planar graph on which a robber can indefinitely evade three cops in the one-cop-moves game. This answers a question recently raised by Sullivan, Townsend and Werzanski.


  1. 1.
    Aigner, M., Fromme, M.: A game of cops and robbers. Discrete Appl. Math. 8, 1–12 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bal, D., Bonato, A., Kinnersley, W.B., Pralat, P.: Lazy cops and robbers on hypercubes. Comb. Probab. Comput. 24, 829–837 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bal, D., Bonato, A., Kinnersley, W.B., Pralat, P.: Lazy cops and robbers played on random graphs and graphs on surfaces. Int. J. Comb. 7, 627–642 (2016)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bonato, A., Nowakowski, R.: The Game of Cops and Robbers on Graphs. American Mathematical Society, Providence (2011)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bonato, A.: Conjectures on cops and robbers. In: Gera, R., Hedetniemi, S., Larson, C. (eds.) Graph Theory. PBM, pp. 31–42. Springer, Cham (2016). CrossRefGoogle Scholar
  6. 6.
    Chung, T.H., Hollinger, G.A., Isler, V.: Search and pursuit-evasion in mobile robotics. Auton. Robots 31, 299–316 (2011)CrossRefGoogle Scholar
  7. 7.
    Clarke, N.E., MacGillivray, G.: Characterizations of k-copwin graphs. Discrete Math. 312, 1421–1425 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Gao, Z., Yang, B.: The cop number of the one-cop-moves game on planar graphs. Preprint.
  9. 9.
    Isaza, A., Lu, J., Bulitko, V., Greiner, R.: A cover-based approach to multi-agent moving target pursuit. In: Proceedings of the 4th Conference on Artificial Intelligence and Interactive Digital Entertainment, pp. 54–59 (2008)Google Scholar
  10. 10.
    Lipton, R., Tarjan, R.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 177–189 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Loh, P., Oh, S.: Cops and robbers on planar directed graphs. J. Graph Theory 86, 329–340 (2017)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Moldenhauer, C., Sturtevant, N.R.: Evaluating strategies for running from the cops. In: Proceedings of the 21st International Joint Conference on Artificial intelligence, IJCAI 2009, pp. 584–589 (2009)Google Scholar
  13. 13.
    Nowakowski, R., Winkler, P.: Vertex to vertex pursuit in a graph. Discrete Math. 43, 235–239 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Offner, D., Okajian, K.: Variations of cops and robber on the hypercube. Australas. J. Comb. 59(2), 229–250 (2014)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Quilliot, A.: Jeux et pointes fixes sur les graphes. Thèse de 3ème cycle, Universit de Paris VI, pp. 131–145 (1978)Google Scholar
  16. 16.
    Sim, K.A., Tan, T.S., Wong, K.B.: Lazy cops and robbers on generalized hypercubes. Discrete Math. 340, 1693–1704 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Simard, F., Morin, M., Quimper, C.-G., Laviolette, F., Desharnais, J.: Bounding an optimal search path with a game of cop and robber on graphs. In: Pesant, G. (ed.) CP 2015. LNCS, vol. 9255, pp. 403–418. Springer, Cham (2015). Google Scholar
  18. 18.
    Sullivan, B.W., Townsend, N., Werzanski, M.: The \(3 \times 3\) rooks graph is the unique smallest graph with lazy cop number 3. Preprint.
  19. 19.
    West, D.B.: Introduction to Graph Theory. Prentice Hall, Upper Saddle River (2000)Google Scholar
  20. 20.
    Yang, B., Hamilton, W.: The optimal capture time of the one-cop-moves game. Theoret. Comput. Sci. 588, 96–113 (2015)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of ReginaReginaCanada

Personalised recommendations