On the decomposition of connected graphs into their biconnected components

TL;DR

This paper presents a recursive method for generating equivalence classes of biconnected graphs, extending to connected graphs, with algorithms suitable for computer implementation.

Contribution

It introduces a recursion formula and linear map for decomposing connected graphs into biconnected components, enhancing graph generation techniques.

Findings

Provides a recursive formula for biconnected graph classes

Extends results to connected and 2-edge connected graphs

Algorithms are suitable for computer implementation

Abstract

We give a recursion formula to generate all equivalence classes of biconnected graphs with coefficients given by the inverses of the orders of their groups of automorphisms. We give a linear map to produce a connected graph with say, u, biconnected components from one with u-1 biconnected components. We use such map to extend the aforesaid result to connected or 2-edge connected graphs. The underlying algorithms are amenable to computer implementation.