Cograph generation with linear delay

TL;DR

This paper introduces a linear-delay algorithm for generating all unlabeled cographs with n vertices efficiently, avoiding duplicates and providing a linear ordering of the generated graphs.

Contribution

The authors present a novel algorithm that generates all unlabeled cographs with linear delay, improving efficiency and avoiding isomorphic duplicates.

Findings

Generation delay is O(n) per cograph.

Overall complexity is O(n * M_n), where M_n is the number of cographs.

Produces a linear ordering of unlabeled cographs.

Abstract

Cographs have always been a research target in areas such as coloring, graph decomposition, and spectral theory. In this work, we present an algorithm to generate all unlabeled cographs with $n$ vertices, based on the generation of cotrees. The delay of our algorithm (time spent between two consecutive outputs) is $O(n)$. The time needed to generate the first output is also $O(n)$, which gives an overall $O(n\,M_n)$ time complexity, where $M_n$ is the number of unlabeled cographs with $n$ vertices. The algorithm avoids the generation of duplicates (isomorphic outputs) and produces, as a by-product, a linear ordering of unlabeled cographs wih $n$ vertices.