Browse > Article
http://dx.doi.org/10.4134/BKMS.b190097

TWO REMARKS ON THE GAME OF COPS AND ROBBERS  

Shitov, Yaroslav (Moscow Institute of Physics and Technology)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.1, 2020 , pp. 127-131 More about this Journal
Abstract
We discuss two unrelated topics regarding Cops and Robbers, a well-known pursuit-evasion game played on a simple graph. First, we address a recent question of Breen et al. and prove the PSPACE-completeness of the cop throttling number, that is, the minimal possible sum of the number k of cops and the number capt(k) of moves that the robber can survive against k cops under the optimal play of both sides. Secondly, we revisit a teleporting version of the game due to Wagner; we disprove one of his conjectures and suggest a new related research problem.
Keywords
Graph theory; cops and robbers;
Citations & Related Records
연도 인용수 순위
  • Reference
1 M. Aigner and M. Fromme, A game of cops and robbers, Discrete Appl. Math. 8 (1984), no. 1, 1-11. https://doi.org/10.1016/0166-218X(84)90073-8   DOI
2 A. Bonato and A. Burgess, Cops and robbers on graphs based on designs, J. Combin. Des. 21 (2013), no. 9, 404-418. https://doi.org/10.1002/jcd.21331   DOI
3 A. Bonato, P. Golovach, G. Hahn, and J. Kratochvil, The capture time of a graph, Discrete Math. 309 (2009), no. 18, 5588-5595. https://doi.org/10.1016/j.disc.2008.04.004   DOI
4 A. Bonato, X. Perez-Gimenez, P. Pralat, and B. Reiniger, The game of overprescribed cops and robbers played on graphs, Graphs Combin. 33 (2017), no. 4, 801-815. https://doi.org/10.1007/s00373-017-1815-2   DOI
5 J. Breen, B. Brimkov, J. Carlson, L. Hogben, K. E. Perry, and C. Reinhart, Throttling for the game of Cops and Robbers on graphs, Discrete Math. 341 (2018), no. 9, 2418-2430. https://doi.org/10.1016/j.disc.2018.05.017   DOI
6 R. G. Downey and M. R. Fellows, Parameterized complexity, Monographs in Computer Science, Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1- 4612-0515-9
7 J. Flum and M. Grohe, Parameterized Complexity Theory, Texts in Theoretical Computer Science. An EATCS Series, Springer-Verlag, Berlin, 2006.
8 F. V. Fomin, P. A. Golovach, J. Kratochvil, N. Nisse, and K. Suchan, Pursuing a fast robber on a graph, Theoret. Comput. Sci. 411 (2010), no. 7-9, 1167-1181. https://doi.org/10.1016/j.tcs.2009.12.010   DOI
9 F. V. Fomin, P. A. Golovach, and P. Pralat, Cops and robber with constraints, SIAM J. Discrete Math. 26 (2012), no. 2, 571-590. https://doi.org/10.1137/110837759   DOI
10 A. S. Fraenkel, M. R. Garey, D. S. Johnson, T. Schaefer, and Y. Yesha, The complexity of checkers on an n$\times$n board, 19th Annual Symposium on Foundations of Computer Science, 55-64, IEEE, 1978.
11 W. B. Kinnersley, Cops and robbers is EXPTIME-complete, J. Combin. Theory Ser. B 111 (2015), 201-220. https://doi.org/10.1016/j.jctb.2014.11.002   DOI
12 A. S. Fraenkel and D. Lichtenstein, Computing a perfect strategy for $n{\times}n$ chess requires time exponential in n, J. Combin. Theory Ser. A 31 (1981), no. 2, 199-214. https://doi.org/10.1016/0097-3165(81)90016-9   DOI
13 A. S. Goldstein and E. M. Reingold, The complexity of pursuit on a graph, Theoret. Comput. Sci. 143 (1995), no. 1, 93-112. https://doi.org/10.1016/0304-3975(95)80012-3   DOI
14 S. Iwata and T. Kasai, The Othello game on an n$\times$n board is PSPACE-complete, Theoret. Comput. Sci. 123 (1994), no. 2, 329-340. https://doi.org/10.1016/0304-3975(94)90131-7   DOI
15 W. B. Kinnersley, Bounds on the length of a game of Cops and Robbers, Discrete Math. 341 (2018), no. 9, 2508-2518. https://doi.org/10.1016/j.disc.2018.05.025   DOI
16 M. Mamino, On the computational complexity of a game of cops and robbers, Theoret. Comput. Sci. 477 (2013), 48-56. https://doi.org/10.1016/j.tcs.2012.11.041   DOI
17 S. Neufeld and R. Nowakowski, A game of cops and robbers played on products of graphs, Discrete Math. 186 (1998), no. 1-3, 253-268. https://doi.org/10.1016/S0012-365X(97)00165-9   DOI
18 A. Quilliot, Points fixes, retractions sur des structures discretes et applications, in Seminaire d'Analyse, 1987-1988 (Clermont-Ferrand, 1987-1988), Exp. 2, 10 pp, Univ. Clermont-Ferrand II, Clermont, 1990.
19 R. Nowakowski and P. Winkler, Vertex-to-vertex pursuit in a graph, Discrete Math. 43 (1983), no. 2-3, 235-239. https://doi.org/10.1016/0012-365X(83)90160-7   DOI
20 S. Reisch, Gobang ist PSPACE-vollstandig, Acta Inform. 13 (1980), no. 1, 59-66.   DOI
21 Z. A. Wagner, Cops and Robbers on diameter two graphs, Discrete Math. 338 (2015), no. 3, 107-109. https://doi.org/10.1016/j.disc.2014.10.012   DOI