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  • Cited by 4
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    McDiarmid, Colin and Scott, Alex 2016. Random graphs from a block-stable class. European Journal of Combinatorics, Vol. 58, p. 96.


    Drmota, M. Gimenez, O. Noy, M. Panagiotou, K. and Steger, A. 2014. The maximum degree of random planar graphs. Proceedings of the London Mathematical Society, Vol. 109, Issue. 4, p. 892.


    Giménez, Omer Noy, Marc and Rué, Juanjo 2013. Graph classes with given 3-connected components: Asymptotic enumeration and random graphs. Random Structures & Algorithms, Vol. 42, Issue. 4, p. 438.


    ADDARIO-BERRY, L. MCDIARMID, C. and REED, B. 2012. Connectivity for Bridge-Addable Monotone Graph Classes. Combinatorics, Probability and Computing, Vol. 21, Issue. 06, p. 803.


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3-Connected Cores In Random Planar Graphs

  • NIKOLAOS FOUNTOULAKIS (a1) and KONSTANTINOS PANAGIOTOU (a1)
  • DOI: http://dx.doi.org/10.1017/S0963548310000532
  • Published online: 24 January 2011
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: (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;

(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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[5]M. Drmota (2009) Random Trees: An Interplay between Combinatorics and Probability, Springer.

[9]O. Giménez and M. Noy (2009) Asymptotic enumeration and limit laws of planar graphs. J. Amer. Math. Soc. 22 309329.

[11]S. Janson , T. Łuczak and A. Ruciński (2000) Random Graphs, Wiley.

[12]C. McDiarmid , A. Steger and D. Welsh (2005) Random planar graphs. J. Combin. Theory Ser. B 93 187205.

[16]T. R. S. Walsh (1982) Counting labelled 3-connected and homeomorphically irreducible 2-connected graphs. J. Combin. Theory Ser. B 32 111.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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