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Enumerating Two-Orbit Graphs

David Seka, Stefan Szeider

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.

Enumerating Two-Orbit Graphs

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.

Paper Structure

This paper contains 10 sections, 3 theorems, 1 equation, 1 table.

Table of Contents

  1. Introduction
  2. Alternative Approaches
  3. From Groups to Graphs
  4. Enumerating Relevant Groups
  5. Optimizations
  6. Pruning Essential Kernels
  7. Deduplicating Orbital Configurations
  8. Round 1 (label-preserving).
  9. Round 2 (scheme isomorphism).
  10. Conclusion

Key Result

Lemma 1

Let $G$ be a graph and $a \in \mathrm{Aut}(G)$. Then $\{v,w\} \in E(G)$ if and only if $\{a(v), a(w)\} \in E(G)$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (7)

  • Lemma 1
  • proof
  • Definition 2
  • Definition 3
  • Theorem 4: Goursat's Lemma lang
  • Proposition 5
  • proof