Generating connected and biconnected graphs

TL;DR

This paper revises an algebraic algorithm for generating connected graphs and extends it to generate biconnected, simple, and loopless graphs using basic graph transformations, improving efficiency and scope.

Contribution

It revises the existing algebraic formula for connected graphs and extends it to broader classes of graphs with a graph transformation-based approach.

Findings

Efficient recursive algorithm for generating connected graphs.

Extension of the algorithm to biconnected, simple, and loopless graphs.

Method relies solely on basic graph transformations.

Abstract

We focus on the algorithm underlying the main result of [A. Mestre, R. Oeckl, Generating loop graphs via Hopf algebra in quantum field theory. J. Math. Phys., 47, 122302, 2006]. This is an algebraic formula to generate all connected graphs in a recursive and efficient manner. The key feature is that each graph carries a scalar factor given by the inverse of the order of its group of automorphisms. In the present paper, we revise that algorithm on the level of graphs. Moreover, we extend the result subsequently to further classes of connected graphs, namely, (edge) biconnected, simple and loopless graphs. Our method consists of basic graph transformations only.