ISAAC 2013: Algorithms and Computation pp 174-184

# Cops and Robbers on Intersection Graphs

• Tomás Gavenčiak
• Vít Jelínek
• Pavel Klavík
• Jan Kratochvíl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)

## Abstract

The game of cops and robber, introduced by Nowakowski and Winkler in 1983, is played by two players on a graph G, one controlling k cops and the other one robber, all positioned on VG. The players alternate in moving their pieces to distance at most 1 each. The cops win if they capture the robber, the robber wins by escaping indefinitely. The cop-number of G, that is the smallest k such that k cops win the game, has recently been a widely studied parameter.

Intersection graph classes are defined by their geometric representations: the vertices are represented by certain geometrical shapes and two vertices are adjacent if and only if their representations intersect. Some well-known intersection classes include interval and string graphs. Various properties of many of these classes have been studied recently, including an interest in their game-theoretic properties.

In this paper we show an upper bound on the cop-number of string graphs and sharp bounds on the cop-number of interval filament graphs, circular graphs, circular arc graphs and function graphs. These results also imply polynomial algorithms determining cop-number for all these classes and their sub-classes.

### Keywords

intersection graphs string graphs interval filament graphs cop and robber pursuit games games on graphs

## Preview

Unable to display preview. Download preview PDF.

### 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 (2011)Google Scholar
3. 3.
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)
4. 4.
Gavril, F.: Maximum weight independent sets and cliques in intersection graphs of filaments. Inf. Process. Lett. 73(5-6), 181–188 (2000)
5. 5.
Kratochvíl, J., Goljan, M., Kučera, P.: String graphs. Rozpravy ČSAV.: Řada matem. a přírodních věd, Academia (1986)Google Scholar
6. 6.
Kratochvíl, J.: String graphs. II. recognizing string graphs is NP-hard. Journal of Combinatorial Theory, Series B 52(1), 67–78 (1991)
7. 7.
Mamino, M.: On the computational complexity of a game of cops and robbers. Theor. Comput. Sci. 477, 48–56 (2013)
8. 8.
McKee, T., McMorris, F.: Topics in Intersection Graph Theory. Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (1999)Google Scholar
9. 9.
Nowakowski, R., Winkler, P.: Vertex-to-vertex pursuit in a graph. Discrete Math. 43, 235–239 (1983)
10. 10.
Pergel, M.: Recognition of polygon-circle graphs and graphs of interval filaments is NP-complete. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 238–247. Springer, Heidelberg (2007)
11. 11.
Quilliot, A.: Some results about pursuit games on metric spaces obtained through graph theory techniques. European J. Combin. 7, 55–66 (1986)
12. 12.
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
13. 13.
Sinden, F.W.: Topology of thin film RC-circuits. Bell System Technical Journal 45, 1639–1662 (1966)

## Copyright information

© Springer-Verlag Berlin Heidelberg 2013

## Authors and Affiliations

• Tomás Gavenčiak
• 1
• Vít Jelínek
• 2
• Pavel Klavík
• 2
• Jan Kratochvíl
• 1
1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
2. 2.Computer Science Institute, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic