Graph classes with given 3-connected components: asymptotic enumeration and random graphs

TL;DR

This paper develops a general framework for asymptotic enumeration and limit laws of graph classes with specified 3-connected components, extending previous results to broader classes like planar and series-parallel graphs.

Contribution

It introduces a unified singularity analysis approach for these graph classes, providing new asymptotic formulas and limit laws for various parameters.

Findings

Asymptotic number of graphs derived from singularities of generating functions

Limit laws for parameters like edges, blocks, and components identified as normal or Poisson

A dichotomy in the size of the largest block and 3-connected component, depending on class similarity to planar or series-parallel graphs

Abstract

Consider a family $\mathcal{T}$ of 3-connected graphs of moderate growth, and let $\mathcal{G}$ be the class of graphs whose 3-connected components are graphs in $\mathcal{T}$. We present a general framework for analyzing such graphs classes based on singularity analysis of generating functions, which generalizes previously studied cases such as planar graphs and series-parallel graphs. We provide a general result for the asymptotic number of graphs in $\mathcal{G}$, based on the singularities of the exponential generating function associated to $\mathcal{T}$. We derive limit laws, which are either normal or Poisson, for several basic parameters, including the number of edges, number of blocks and number of components. For the size of the largest block we find a fundamental dichotomy: classes similar to planar graphs have almost surely a unique block of linear size, while classes similar to series-parallel graphs have only sublinear blocks. This dichotomy also applies to the size of the largest 3-connected component. For some classes under study both regimes occur, because of a critical phenomenon as the edge density in the class varies.