AI 中文总结
该研究定义非对称深度以探究图的局部对称性，证明平面图非对称深度的紧上界，发现IPR富勒烯对偶图在47顶点时达极值，揭示部分非对称IPR富勒烯存在自同构群无法察觉的近镜像破缺对称性，并将该深度界推广至高亏格图。
AI 中文摘要
尽管几乎所有图都是非对称的——即不存在非平凡全局自同构——但它们仍可能拥有以导出子图间同构（即部分自同构）形式存在的局部对称性。我们通过非对称深度来研究这类局部对称性，其定义为非平凡部分自同构的最大秩。我们证明了平面图类中非对称深度的紧上界，并确定了极值图：IPR富勒烯的对偶图在仅47个顶点时就达到了最大值。我们的主要结构结果涉及既非最大非对称也非完全对称的IPR富勒烯。在这类笼形结构中，没有纯粹的局部作用能实现低非对称深度，且我们证明，能实现该深度的映射也无法被限制在笼形结构的小部分区域内：既不能限制在单个面，也不能限制在亏格k≤3的界面后方（界面边数最多为5−k）。因此，非对称深度为2或3的笼形结构并非在某一处非对称；它带有自同构群无法察觉的破缺对称性。这类笼形结构十分罕见——在n=118时，非对称IPR富勒烯中占比不到2%。在所有727个这类笼形结构中，最大的部分自同构是近镜像反射，我们将此表述为一个明确的猜想。我们还将非对称深度的上界推广到了更高亏格的图。
英文摘要
Although almost all graphs are asymmetric -- having no nontrivial global automorphisms -- they may still possess local symmetries in the form of isomorphisms between induced subgraphs, i.e., partial automorphisms. We study such local symmetries via asymmetric depth, defined in terms of the maximum rank of a nontrivial partial automorphism. We prove a tight upper bound on asymmetric depth in the class of planar graphs and identify the extremal graphs: duals of IPR fullerenes attain the maximum already on $47$ vertices. Our main structural result concerns the IPR fullerenes that are neither maximally asymmetric nor symmetric. In such a cage no purely local action realises a low asymmetric depth, and we show that the map which does realise it cannot be confined to a small part of the cage either: neither to a single face, nor behind an interface of at most $5-k$ edges, $k \le 3$ being the deficiency. A cage of asymmetric depth $2$ or $3$ is therefore not asymmetric in one place; it carries a broken symmetry invisible to its automorphism group. Such cages are rare -- under $2\%$ of the asymmetric IPR fullerenes at $n = 118$. In all $727$ of them the largest partial automorphism is a near-mirror reflection, which we state as an explicit conjecture. We also extend the asymmetric depth bound to graphs of higher genus.
Comments21 pages, 7 figures, 1 table. Includes computer-assisted verification; code available at https://github.com/JanPastorek/asym_depth_fullerenes
