ISAAC 2015: Algorithms and Computation pp 355-366

# Cops and Robbers on String Graphs

• Tomáš Gavenčiak
• Przemysław Gordinowicz
• Vít Jelínek
• Pavel Klavík
• Jan Kratochvíl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)

## Abstract

The game of cops and robber, introduced by Nowakowski and Winkler in 1983, is played by two players on a graph. One controls k cops and the other a robber. The players alternate and move their pieces to the distance at most one. The cops win if they capture the robber, the robber wins by escaping indefinitely. The cop number of G is the smallest k such that k cops win the game.

We extend the results of Gavenčiak et al. [ISAAC 2013], investigating the maximum cop number of geometric intersection graphs. Our main result shows that the maximum cop number of string graphs is at most 15, improving the previous bound 30. We generalize this approach to string graphs on a surface of genus g to show that the maximum cop number is at most $$10g+15$$, which strengthens the result of Quilliot [J. Combin. Theory Ser. B 38, 89–92 (1985)]. For outer string graphs, we show that the maximum cop number is between 3 and 4. Our results also imply polynomial-time algorithms determining the cop number for all these graph classes.

### References

1. 1.
Aigner, M., Fromme, M.: Game of cops and robbers. Discrete Appl. Math. 8(1), 1–12 (1984)
2. 2.
Bonato, A., Nowakowski, R.J.: The Game of Cops and Robbers on Graphs. American Mathematical Society, Providence (2011)
3. 3.
Esperet, L., Joret, G.: Boxicity of graphs on surfaces. Graphs and Combinatorics 29(3), 417–427 (2013)
4. 4.
Fomin, F.V., Golovach, P.A., Kratochvíl, J., Nisse, N., Suchan, K.: Pursuing a fast robber on a graph. Theor. Comput. Sci. 411(7–9), 1167–1181 (2010)
5. 5.
Gavenčiak, T., Gordinowicz, P., Jelínek, V., Klavík, P., Kratochvíl, J.: Cops and robbers of intersection graphs (in preparation, 2015)Google Scholar
6. 6.
Gavenčiak, Tomás, Jelínek, Vít, Klavík, Pavel, Kratochvíl, Jan: Cops and robbers on intersection graphs. In: Cai, Leizhen, Cheng, Siu-Wing, Lam, Tak-Wah (eds.) Algorithms and Computation. LNCS, vol. 8283, pp. 174–184. Springer, Heidelberg (2013)
7. 7.
Goldstein, A.S., Reingold, E.M.: The complexity of pursuit on a graph. Theor. Comput. Sci. 143(1), 93–112 (1995)
8. 8.
Kinnersley, W.B.: Cops and robbers is exptime-complete. J. Comb. Theor. Ser. B 111, 201–220 (2015)
9. 9.
Kratochvíl, J.: String graphs. II. recognizing string graphs is NP-hard. J. Comb. Theor. Ser. B 52(1), 67–78 (1991)
10. 10.
Mamino, M.: On the computational complexity of a game of cops and robbers. Theor. Comput. Sci. 477, 48–56 (2013)
11. 11.
Mohar, B., Thomassen, C.: Graphs on Surfaces. The John Hopkins University Press, Baltimore (2001)
12. 12.
Nowakowski, R., Winkler, P.: Vertex-to-vertex pursuit in a graph. Discrete Math. 43, 235–239 (1983)
13. 13.
Prasolov, V.: Elements of Combinatorial and Differential Topology. Graduate Studies in Mathematics. American Mathematical Soc., Providence (2006)
14. 14.
Quilliot, A.: A short note about pursuit games played on a graph with a given genus. J. Combin. Theory Ser. B 38, 89–92 (1985)
15. 15.
Schroeder, B.S.W.: The copnumber of a graph is bounded by 3/2 genus(g) + 3. Trends Math., pp. 243–263. Birkhäuser, Boston (2001)Google Scholar

## Authors and Affiliations

• Tomáš Gavenčiak
• 1
• Przemysław Gordinowicz
• 2
• Vít Jelínek
• 3
• Pavel Klavík
• 3
• Jan Kratochvíl
• 1
1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic
2. 2.Institute of MathematicsTechnical University of LodzŁódźPoland
3. 3.Computer Science InstituteCharles University in PraguePragueCzech Republic