Maximal Biconnected Subgraphs of Random Planar Graphs

Abstract

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( ; Pn) be the number of blocks (i.e. maximal biconnected subgraphs) of Pn that contain exactly 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

The maximum degree of random planar graphs

“…It remains to show the claimed upper bound for random (not necessarily) connected planar graphs in Theorem 1.1. To this end, we use the following property, see [19,24]. Let P n be a random planar graph with n vertices, and let ω n be an arbitrary slowly growing function.…”

“…The proof of the lower bound for random connected planar graphs in Theorem 1.1 follows directly from the lower bound in the previous section. More precisely, in [20,24] it was shown that a random planar graph contains with probability 1 − o(1) a very large 2-connected subgraph.…”

“…Boltzmann samplers have proved useful not only for random generation, but also for the analysis of random combinatorial objects. This approach was started in [26] and later pursued in [2,3] and in [16,24,25]. In the present paper, Boltzmann samplers also play a key role.…”

“…In the present paper, Boltzmann samplers also play a key role. A crucial fact, proved independently using probabilistic [24] and analytic methods [20], is that w.h.p. a connected random planar graph has a unique block (2-connected component) of linear size, and the remaining blocks are of order at most n 2/3 .…”

“…We find then ourselves in an unexpected (and satisfactory) situation: analytic methods yield only the upper bound and probabilistic methods yield only the lower bound, with the same multiplicative constant. This can be considered as the culmination of two parallel and independent approaches for analyzing random planar graphs, one based on generating functions and analytic methods [10-12, 19, 20], the other one based on Boltzmann samplers and concentration inequalities [2,3,16,[24][25][26].…”