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 either there is a unique giant core in Bn, that is, there is a 0 < c = c( 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.
either there is a unique giant core in Bn, that is, there is a 0 < c = c(
or all cores of Bn contain O(logn) vertices.
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