Largest 4-connected components of 3-connected planar triangulations
Random Structures & Algorithms
Random maps, coalescing saddles, singularity analysis, and airy phenomena
Random Structures & Algorithms - Special issue on analysis of algorithms dedicated to Don Knuth on the occasion of his (100)8th birthday
The Size of the Largest Components in Random Planar Maps
SIAM Journal on Discrete Mathematics
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
Journal of Combinatorial Theory Series B
On the maximum degree of a random planar graph
Combinatorics, Probability and Computing
Analytic Combinatorics
On degrees in random triangulations of point sets
Journal of Combinatorial Theory Series A
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Let C be a class of labeled connected graphs, and let Cn be a graph drawn uniformly at random from graphs in C that contain exactly n vertices. Denote by b(ℓ; Cn) the number of blocks (i.e., maximal biconnected subgraphs) of Cn that contain exactly ℓ vertices, and let lb(Cn) be the number of vertices in a largest block of Cn. We show that under certain general assumptions on C, Cn belongs with high probability to one of the following categories: (1) lb(Cn) ∼ cn, for some explicitly given c = c(C), and the second largest block is of order nα, where 1 α = α(C), or (2) lb(Cn) = O(log n), that is, all blocks contain at most logarithmically many vertices. Moreover, in both cases we show that the quantity b(ℓ; Cn) is concentrated for all ℓ and we determine its expected value. As a corollary we obtain that the class of planar graphs belongs to category (1). In contrast to that, outerplanar and series-parallel graphs belong to category (2).