Combinatorics, Probability and Computing

Paper

3-Connected Cores In Random Planar Graphs

NIKOLAOS FOUNTOULAKISa1 and KONSTANTINOS PANAGIOTOUa1

a1 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)

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.

(Received July 17 2009)

(Revised December 06 2010)

(Online publication January 24 2011)