Enumerating Two-Orbit Graphs

TL;DR

This paper introduces a novel method for enumerating all connected two-orbit graphs up to 27 vertices, significantly advancing the enumeration capabilities beyond previous methods.

Contribution

The authors develop a new approach leveraging Goursat's lemma and optimizations to efficiently enumerate two-orbit graphs, achieving the first complete enumeration up to 27 vertices.

Findings

Enumerated 10,094,721 connected two-orbit graphs up to 27 vertices.

Developed a scalable enumeration method using group theory and pruning techniques.

Pushed the boundary of graph enumeration beyond existing direct methods.

Abstract

We present an approach to enumerate graphs whose automorphism group has exactly two orbits. Our method exploits the observation that we can enumerate all graphs whose automorphism group contains a given this permutation group. We obtain the relevant groups via Goursat's lemma. In order to scale the enumeration, we employ additional optimizations that prune irrelevant groups. In total, we enumerate, for the first time, all connected two-orbit graphs of up to 27 vertices, totaling 10,094,721 graphs, pushing the state of the art well beyond what direct enumeration methods can achieve.