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3-Connected Cores In Random Planar Graphs

Published online by Cambridge University Press:  24 January 2011

NIKOLAOS FOUNTOULAKIS  and
KONSTANTINOS PANAGIOTOU
NIKOLAOS FOUNTOULAKIS
Affiliation:
Max-Planck-Institute for Informatics, Campus E.1 4, D-66123 Saarbrücken, Germany (e-mail: fountoul@mpi-inf.mpg.de, kpanagio@mpi-inf.mpg.de)
KONSTANTINOS PANAGIOTOU
Affiliation:
Max-Planck-Institute for Informatics, Campus E.1 4, D-66123 Saarbrücken, Germany (e-mail: fountoul@mpi-inf.mpg.de, kpanagio@mpi-inf.mpg.de)
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Abstract

The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints.

In this paper we focus on the structure of random 2-connected planar graphs regarding the sizes of their 3-connected building blocks, which we call cores. In fact, we prove a general theorem regarding random biconnected graphs from various classes. If Bn is a graph drawn uniformly at random from a suitable class of labelled biconnected graphs, then we show that with probability 1 − o(1) as n → ∞, Bn belongs to exactly one of the following categories:

  1. (i) either there is a unique giant core in Bn, that is, there is a 0 < c = c() < 1 such that the largest core contains ~ cn vertices, and every other core contains at most nα vertices, where 0 < α = α() < 1;

  2. (ii) or all cores of Bn contain O(logn) vertices.

Moreover, we find the critical condition that determines the category to which Bn belongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 and n. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particular c = 0.765. . . and α = 2/3.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 3 , May 2011 , pp. 381 - 412
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Bender, E. A., Gao, Z. and Wormald, N. C. (2002) The number of 2-connected planar graphs. Electron. J. Combin. 9 #43.CrossRefGoogle Scholar
[2]Bernasconi, N., Panagiotou, K. and Steger, A. (2008) On properties of random dissections and triangulations. In Proc. 19th Annual ACM–SIAM Symposium on Discrete Algorithms (SODA '08), pp. 132–141.Google Scholar
[3]Bodirsky, M., Giménez, O., Kang, M. and Noy, M. (2005) On the number of series parallel and outerplanar graphs. In 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Vol. AE of DMTCS Proceedings, pp. 383–388.CrossRefGoogle Scholar
[4]Denise, A., Vasconcellos, M. and Welsh, D. J. A. (1996) The random planar graph. Congress. Numer. 113 61–79.Google Scholar
[5]Drmota, M. (2009) Random Trees: An Interplay between Combinatorics and Probability, Springer.CrossRefGoogle Scholar
[6]Duchon, P., Flajolet, P., Louchard, G. and Schaeffer, G. (2004) Boltzmann samplers for the random generation of combinatorial structures. Combin. Probab. Comput. 13 577–625.CrossRefGoogle Scholar
[7]Flajolet, F. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
[8]Gerke, S., Giménez, O. and Weissl, A. (2008) On the number of graphs not containing K 3,3 as a minor. Electron. J. Combin. 15 R114.CrossRefGoogle Scholar
[9]Giménez, O. and Noy, M. (2009) Asymptotic enumeration and limit laws of planar graphs. J. Amer. Math. Soc. 22 309–329.CrossRefGoogle Scholar
[10]Giménez, O., Noy, M. and Rué, J. (2009) Graph classes with given 3-connected components: asymptotic enumeration and random graphs. Manuscript, available at: http://arxiv.org/abs/0907.0376.Google Scholar
[11]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.CrossRefGoogle Scholar
[12]McDiarmid, C., Steger, A. and Welsh, D. (2005) Random planar graphs. J. Combin. Theory Ser. B 93 187–205.CrossRefGoogle Scholar
[13]Panagiotou, K. and Steger, A. (2009) Maximal biconnected subgraphs of random planar graphs. In Proc. 20th Annual ACM–SIAM Symposium on Discrete Algorithms (SODA '09), pp. 432–440.CrossRefGoogle Scholar
[14]Trakhtenbrot, B. A. (1958) Towards a theory of non-repeating contact schemes. Trudi Mat. Inst. Akad. Nauk SSSR 51 226–269.Google Scholar
[15]Tutte, W. T. (1966) Connectivity in Graphs, University of Toronto Press.CrossRefGoogle Scholar
[16]Walsh, T. R. S. (1982) Counting labelled 3-connected and homeomorphically irreducible 2-connected graphs. J. Combin. Theory Ser. B 32 1–11.CrossRefGoogle Scholar