  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 biconnected 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. If B_n is a graph
drawn uniformly at random from a class B of labeled biconnected graphs, then we
show that with probability 1-o(1) B_n belongs to exactly one of the following
categories:
  (i) Either there is a unique giant core in B_n, that is, there is a 0 < c < 1
such that the largest core contains ~ cn vertices, and every other core
contains at most n^a vertices, where 0 < a < 1; (ii) or all cores of B_n
contain O(log n) vertices.
  Moreover, we find the critical condition that determines the category to
which B_n 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 a = 2/3.
