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.

翻译：我们提出一种枚举自同构群恰好具有两个轨道的图的方法。该方法利用了以下观察：我们可以枚举所有自同构群包含给定置换群的图。通过高萨特引理获得相关群。为扩大枚举规模，我们采用额外优化来剪枝无关群。最终，我们首次枚举了所有顶点数不超过27的连通双轨道图，共计10,094,721个图，将现有技术水平提升至远超直接枚举方法所能达到的范畴。