3-Connected Cores In Random Planar Graphs

TL;DR

This paper investigates the structure of large random biconnected planar graphs, revealing a phase transition in core sizes and establishing that such graphs typically contain a unique giant 3-connected core with high probability.

Contribution

It provides a general theorem characterizing the size distribution of cores in random biconnected graphs, especially planar ones, and identifies conditions for the emergence of a giant core.

Findings

A giant core contains approximately 76.5% of vertices in typical planar graphs.

Cores are either dominated by a large core or are all logarithmic in size.

Sharp concentration results for core counts of all sizes are established.

Abstract

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 biconnected 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. If B_n is a graph drawn uniformly at random from a class B of labeled biconnected graphs, then we show that with probability 1-o(1) B_n belongs to exactly one of the following categories: (i) Either there is a unique giant core in B_n, that is, there is a 0 < c < 1 such that the largest core contains ~ cn vertices, and every other core contains at most n^a vertices, where 0 < a < 1; (ii) or all cores of B_n contain O(log n) vertices. Moreover, we find the critical condition that determines the category to which B_n 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 a = 2/3.