Enumerating Two-Orbit Graphs

2026-04-01

AI Summary

This study addresses the efficient enumeration of connected graphs whose automorphism groups act with exactly two orbits. To this end, the authors propose a novel methodology that integrates Goursat’s lemma to construct candidate groups, performs graph enumeration under automorphism group constraints, and incorporates group-theoretic pruning to enhance computational efficiency. This approach yields the first complete enumeration of all connected two-orbit graphs on up to 27 vertices, resulting in a total of 10,094,721 such graphs. The method substantially surpasses the limitations of traditional brute-force enumeration techniques, dramatically expanding the scale of instances that can be feasibly solved within this class of symmetry-constrained graph enumeration problems.

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.

Problem

• two-orbit graphs
• graph enumeration
• automorphism group
• graph orbits

Innovation

• two-orbit graphs
• automorphism group
• Goursat's lemma
• graph enumeration
• group pruning