AI 中文总结
该研究针对五类多面体图极值族，证明含各度数顶点的最小阶3-多面体图不对称，确定补图为多面体的三个图的自同构群，分类半径1、自对偶单图及图乘积多面体的自同构群。
AI 中文摘要
我们研究五类多面体图极值族中的自同构群。对所有n≥14，我们证明每个包含度为3、4、…、n各一个顶点的最小阶3-多面体图都是不对称的。该证明采用精确平面亏格分解、高度顶点尾部的完整刻画，以及由唯一高度顶点诱导子图的饱和定理。对偶性给出包含面为3、4、…、n各一个面的最小面多面体的对应不对称结果。对于补图也为多面体的三个多面体图，我们确定其普通自同构群和扩展自同构群，并证明扩展群Aut⁺(G₁₃)≅(C₂×C₂)⋊C₄。接下来，我们对半径为1的多面体的自同构群进行分类：在唯一支配顶点情形下为循环群或二面体群，三角剖分情形下的可能群为1、C₂、C₃、C₂×C₂、S₃。对于自对偶类中的单图多面体，我们证明其自同构群为1或C₂。最后，我们考虑图乘积多面体，针对四类标准图乘积，按自同构群对其进行分类。
英文摘要
We study automorphism groups in five extremal families of polyhedral graphs. For every $n\ge14$, we prove that every minimum-order $3$-polytopal graph containing a vertex of each degree $3,4,\ldots,n$ is asymmetric. The proof uses an exact planar defect decomposition, a complete description of the high-degree tail, and a saturation theorem for the subgraph induced by the uniquely high-degree vertices. Duality gives the corresponding asymmetry result for minimum-face polyhedra containing faces of every size $3,4,\ldots,n$. For the three polyhedral graphs whose complements are also polyhedral, we determine the ordinary and extended automorphism groups and identify the extended group \[ \mathsf{Aut}^{\pm}(G_{13})\cong (C_2\times C_2)\rtimes C_4. \] Next, we classify automorphism groups of radius-one polyhedra. In the unique-dominating-vertex case they are cyclic or dihedral, and in the triangulated case the possibilities are \[ 1,\qquad C_2,\qquad C_3,\qquad C_2\times C_2,\qquad S_3. \] For polyhedra that are unigraphic among the class of self-dual, we show that their automorphism group is either $1$ or $C_2$. Finally, we consider polyhedra that are products of graphs, for each of the four standard graph products, and we classify them according to their automorphism group.
