A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components

TL;DR

This paper develops a comprehensive grammar framework for decomposing graphs into 3-connected components, extending Tutte's canonical decomposition using symbolic combinatorics and an extended dissymmetry theorem.

Contribution

It translates Tutte's graph decomposition into a symbolic grammar applicable to graph families, incorporating negative signs via an extended dissymmetry theorem.

Findings

Recovered the analytic expression for counting labelled planar graphs.

Provided a combinatorial derivation of planar graph enumeration.

Utilized bijective constructions of planar maps in the methodology.

Abstract

Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general grammar expressing any family of graphs (with some stability conditions) in terms of the 3-connected subfamily. A key ingredient we use is an extension of the so-called dissymmetry theorem, which yields negative signs in the grammar. As a main application we recover in a purely combinatorial way the analytic expression found by Gim\'enez and Noy for the series counting labelled planar graphs (such an expression is crucial to do asymptotic enumeration and to obtain limit laws of various parameters on random planar graphs). Besides the grammar, an important ingredient of our method is a recent bijective construction of planar maps by Bouttier, Di Francesco and Guitter.