Two-connected graphs with prescribed three-connected components

TL;DR

This paper extends the classical 3-decomposition of 2-connected graphs to simple graphs, introducing a bicolored tree structure that relates 2-connected graphs with prescribed 3-connected components, leading to new enumeration formulas.

Contribution

It adapts the 3-decomposition to simple graphs and establishes a bicolored tree framework linking 2-connected graphs with specified 3-connected components, along with related enumeration results.

Findings

Derived a dissymmetry theorem for 2-connected graphs with given 3-connected components.

Established functional equations characterizing classes of two-pole networks.

Provided enumerative formulas for classes of graphs based on their 3-connected components.

Abstract

We adapt the classical 3-decomposition of any 2-connected graph to the case of simple graphs (no loops or multiple edges). By analogy with the block-cutpoint tree of a connected graph, we deduce from this decomposition a bicolored tree tc(g) associated with any 2-connected graph g, whose white vertices are the 3-components of g (3-connected components or polygons) and whose black vertices are bonds linking together these 3-components, arising from separating pairs of vertices of g. Two fundamental relationships on graphs and networks follow from this construction. The first one is a dissymmetry theorem which leads to the expression of the class B=B(F) of 2-connected graphs, all of whose 3-connected components belong to a given class F of 3-connected graphs, in terms of various rootings of B. The second one is a functional equation which characterizes the corresponding class R=R(F) of two-pole networks all of whose 3-connected components are in F. All the rootings of B are then expressed in terms of F and R. There follow corresponding identities for all the associated series, in particular the edge index series. Numerous enumerative consequences are discussed.