Maximal biconnected subgraphs of random planar graphs

• Authors: Konstantinos Panagiotou;Angelika Steger
• Venue: SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
• Year: 2009

Let Pn be the class of simple labeled planar graphs with n vertices, and denote by Pn a graph drawn uniformly at random from this set. Basic properties of Pn were first investigated by Denise, Vasconcellos, and Welsh [7]. Since then, the random planar graph has attracted considerable attention, and is nowadays 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 study closely the structure of Pn. More precisely, let b(l; Pn) be the number of blocks (i.e. maximal biconnected subgraphs) of Pn that contain exactly l vertices, and let lb(Pn) be the number of vertices in the largest block of Pn. We show that with high probability Pn contains a giant block that includes up to lower order terms cn vertices, where c ≈ 0.959 is an analytically given constant. Moreover, we show that the second largest block contains only [EQUATION] vertices, and prove sharp concentration results for b(l; Pn), for all 2 ≤ l ≤ n2/3 (here [EQUATION](.) stands for "up to logarithmic factors"). In fact, we obtain this result as a consequence of a much more general result that we prove in this paper. 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. Under certain assumptions on C, and depending on the behavior of the singularity of the generating function enumerating the elements of C, Cn belongs with high probability to one of the following three categories, which differ vastly in complexity. Cn either (1) behaves like a random planar graph, i.e. lb(Cn) ~ cn, for some analytically 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), i.e., all blocks contain at most logarithmically many vertices, or (3) lb(Cn) = Õ(nα), for some α = α(C) Planar graphs belong to category (1). In contrast to that, outerplanar and series-parallel graphs belong to category (2).