Abstract
We define helicopter cop and robber games with multiple robbers, extending previous research, which only considered the pursuit of a single robber. Our model is defined for robbers that are visible (the cops know their position) and active (able to move at every turn) but is easily adapted to other common variants of the game. The game with many robbers is non-monotone: more cops are needed if their moves are restricted so as to monotonically decrease the space available to the robbers. Because the cops may decide their moves based on the robbers’ current position, strategies in the game are interactive but the game becomes, in a sense, less interactive as the initial number of robbers increases. We prove that the main parameter emerging from the game captures a hierarchy of parameters between proper pathwidth and proper treewidth. We give a complete characterization of the parameter for trees and an upper bound for general graphs.
This research is supported by the Project “CAPODISTRIAS” (AΠ736/24.3.2006) of the National and Capodistrian University of Athens (project code: 70/4/8757).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bienstock, D., Seymour, P.D.: Monotonicity in graph searching. J. Algorithms 12(2), 239–245 (1991)
Colin de Verdière, Y.: Multiplicities of eigenvalues and tree-width of graphs. J. Combin. Theory Ser. B 74(2), 121–146 (1998)
Dendris, N.D., Kirousis, L.M., Thilikos, D.M.: Fugitive-search games on graphs and related parameters. Theoret. Comput. Sci. 172(1–2), 233–254 (1997)
Ellis, J.A., Sudborough, I.H., Turner, J.S.: The vertex separation and search number of a graph. Information and Computation 113(1), 50–79 (1994)
Fomin, F.V., Fraigniaud, P., Nisse, N.: Nondeterministic graph searching: from pathwidth to treewidth. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 364–375. Springer, Heidelberg (2005)
Fomin, F.V., Thilikos, D.M.: Multiple edges matter when searching a graph. Unpublished manuscript
Hunter, P., Kreutzer, S.: Digraph measures: Kelly decompositions, games, and orderings. In: SODA. 18th ACM-SIAM Symp. on Disc. Algorithms, pp. 637–644 (2007)
Seymour, P.D., Thomas, R.: Graph searching and a min-max theorem for tree-width. J. Combin. Theory Ser. B 58(1), 22–33 (1993)
Stamatiou, Y.C., Thilikos, D.M.: Monotonicity and inert fugitive search games. Electronic Notes in Discrete Mathematics 3 (1999)
Takahashi, A., Ueno, S., Kajitani, Y.: Minimal acyclic forbidden minors for the family of graphs with bounded path-width. Disc. Math. 127(1–3), 293–304 (1994)
Takahashi, A., Ueno, S., Kajitani, Y.: Mixed searching and proper-path-width. Theoret. Comput. Sci. 137(2), 253–268 (1995)
Thilikos, D.M.: Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Math. 105, 239–271 (2000)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Richerby, D., Thilikos, D.M. (2007). Graph Searching in a Crime Wave. In: Brandstädt, A., Kratsch, D., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2007. Lecture Notes in Computer Science, vol 4769. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74839-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-74839-7_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74838-0
Online ISBN: 978-3-540-74839-7
eBook Packages: Computer ScienceComputer Science (R0)