Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-25T06:27:41.466Z Has data issue: false hasContentIssue false
  • English
  • Français

Revolutionaries and Spies on Random Graphs

Published online by Cambridge University Press:  08 March 2013

DIETER MITSCHE  and
PAWEŁ PRAŁAT
DIETER MITSCHE
Affiliation:
Department of Mathematics, Ryerson University, Toronto, ON, Canada (e-mail: dmitsche@ryerson.ca, pralat@ryerson.ca)
PAWEŁ PRAŁAT
Affiliation:
Department of Mathematics, Ryerson University, Toronto, ON, Canada (e-mail: dmitsche@ryerson.ca, pralat@ryerson.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a simplified model for network security. In the game we consider in this paper, a team of r revolutionaries tries to hold an unguarded meeting consisting of m revolutionaries. A team of s spies wants to prevent this forever. For given r and m, the minimum number of spies required to win on a graph G is the spy number σ(G,r,m). We present asymptotic results for the game played on random graphs G(n,p) for a large range of p = p(n), r=r(n), and m=m(n). The behaviour of the spy number is analysed completely for dense graphs (that is, graphs with average degree at least n1/2+ε for some ε > 0). For sparser graphs, some bounds are provided.

Keywords

Primary 05C80Secondary 05C5768R10
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 3 , May 2013 , pp. 417 - 432
Copyright
Copyright © Cambridge University Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albert, M. H., Nowakowski, R. J. and Wolfe, D. (2007) Lessons in Play, A. K. Peters.Google Scholar
Alspach, B. (2006) Sweeping and searching in graphs: A brief survey. Matematiche 59 5–37.Google Scholar
Bonato, A. and Nowakowski, R. J. (2011) The Game of Cops and Robbers on Graphs, AMS.CrossRefGoogle Scholar
Bollobás, B., Kun, G. and Leader, I. (2013) Cops and robbers in a random graph. J. Combina. Theory Ser. B 103 (2)226–236.Google Scholar
Bonato, A., Prałat, P. and Wang, C. (2009) Network security in models of complex networks. Internet Mathematics 4 419–436.CrossRefGoogle Scholar
Butterfield, J. V., Cranston, D. W., Puleo, G., West, D. B. and Zamani, R. (2012) Revolutionaries and Spies: Spy-good and spy-bad graphs. Theoret. Comput. Sci. 463 35–53.Google Scholar
Cranston, D. W., Smyth, C. D. and West, D. B. (2012) Revolutionaries and spies in trees and unicyclic graphs. J. Combin. 3 195–206.Google Scholar
Fomin, F. V. and Thilikos, D. (2008) An annotated bibliography on guaranteed graph searching. Theoret. Comput. Sci. 399 236–245.CrossRefGoogle Scholar
Hahn, G. (2007) Cops, robbers and graphs. Tatra Mt Math. Publ. 36 163–176.Google Scholar
Howard, D. and Smyth, C. D. (2012) Revolutionaries And Spies. Discrete Math. 312 (22)3384–3391.Google Scholar
Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.Google Scholar
Łuczak, T. and Prałat, P. (2010) Chasing robbers on random graphs: Zigzag theorem. Random Struct. Alg. 37 516–524.Google Scholar
Nowakowski, R. and Winkler, P. (1983) Vertex-to-vertex pursuit in a graph. Discrete Math. 43 230–239.Google Scholar
Prałat, P. (2010) When does a random graph have constant cop number? Australas. J. Combin. 46 285–296.Google Scholar
Prałat, P. and Wormald, N. Meyniel's conjecture holds for random graphs. Preprint.Google Scholar
Prałat, P. and Wormald, N. Meyniel's conjecture holds for random d-regular graphs. Preprint.Google Scholar
Quilliot, A. (1978) Jeux et pointes fixes sur les graphes. PhD dissertation, Université de Paris VI.Google Scholar