Cops and Robbers on Intersection Graphs
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 V G . 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.
Keywordsintersection graphs string graphs interval filament graphs cop and robber pursuit games games on graphs
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- 2.Bonato, A., Nowakowski, R.J.: The Game of Cops and Robbers on Graphs. American Mathematical Society (2011)Google Scholar
- 5.Kratochvíl, J., Goljan, M., Kučera, P.: String graphs. Rozpravy ČSAV.: Řada matem. a přírodních věd, Academia (1986)Google Scholar
- 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
- 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