Boltzmann Samplers for the Random Generation of Combinatorial Structures
Combinatorics, Probability and Computing
Journal of Combinatorial Theory Series B
On properties of random dissections and triangulations
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Maximal biconnected subgraphs of random planar graphs
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Analytic Combinatorics
Random Trees: An Interplay between Combinatorics and Probability
Random Trees: An Interplay between Combinatorics and Probability
On the degree distribution of random planar graphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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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() cn vertices, and every other core contains at most nα vertices, where 0 ) (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.