伪曲面中图嵌入的对偶性与 minors
Duality and minors for embeddings of graphs in pseudosurfaces
- Belmont University（贝尔蒙特大学）
- Vanderbilt University（范德堡大学）
- University of Amsterdam（阿姆斯特丹大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文定义伪曲面中图的伪胞腔嵌入，允许 pinchpoint 位于非顶点处，从而为对偶性和 minor 运算提供简单定义，并推广了多种现有结构，同时开发了多项式不变量族。
AI中文摘要：
曲面中图的胞腔嵌入具有定义良好的对偶性和 minor（边收缩与删除）运算，这些运算以自然的方式相互作用。伪曲面是通过对曲面（紧致二维流形）进行有限次有限点集的识别而得到的。由这些识别产生的点不具有与开圆盘同胚的邻域，被称为 pinchpoint。自 20 世纪 60 年代以来，人们一直以隐式和显式的方式考虑伪曲面中的图嵌入。通常要求所有 pinchpoint 对应于图的顶点。然而，这使得为伪曲面嵌入定义对偶性和 minor 并让这些运算以预期方式相互作用变得困难。我们定义了伪曲面中图的伪胞腔嵌入类，它允许 pinchpoint 出现在顶点以外的位置，特别是在面或边的中间。一个称为拟胞腔嵌入的子类对应于 Deneen、Shute 和 Thomborson 以及 Huggett 和 Moffatt 先前的嵌入模型。伪胞腔嵌入还推广了其他结构，包括 Ellis-Monaghan、Kauffman 和 Moffatt 的边点带状图，以及循环有序图或 cog（也称为刚性顶点图）。伪胞腔嵌入的对偶性和 minor 运算使用拓扑商运算具有非常简单直接的定义。无边的图的伪胞腔嵌入具有非平凡的结构，我们为这些图定义了一些与二分图上的“t-minor”运算相关的 minor 运算。我们为伪胞腔嵌入开发了一族多项式不变量，并讨论了与其他多项式不变量的联系。
英文摘要：
Cellular embeddings of graphs in surfaces have well-defined duality and minor (edge contraction and deletion) operations that interact in a natural way. A pseudosurface is obtained from a surface (compact 2-manifold) by a finite number of identifications of finite sets of points. Points that are created by the identifications do not have a neighborhood homeomorphic to an open disk and are known as pinchpoints. Embeddings of graphs in pseudosurfaces have been considered, both implicitly and explicitly, since the 1960s. Usually the condition that all pinchpoints correspond to vertices of the graph is imposed. However, this makes it difficult to define duality and minors for pseudosurface embeddings and have these operations interact in the expected way. We define the class of pseudocellular embeddings of graphs in pseudosurfaces, which allow pinchpoints at places other than vertices, in particular in the middle of faces or edges. A subclass known as quasicellular embeddings corresponds to previous embedding models due to Deneen, Shute, and Thomborson and to Huggett and Moffatt. Pseudocellular embeddings also generalize other structures, including the edge-point ribbon graphs of Ellis-Monaghan, Kauffman, and Moffatt, and cyclically ordered graphs or cogs (also known as rigid-vertex graphs). Duality and minor operations for pseudocellular embeddings have very simple and straightforward definitions using topological quotient operations. Pseudocellular embeddings of edgeless graphs have nontrivial structure, and we define some minor operations for those that are related to `t-minor' operations on bipartite graphs. We develop a family of polynomial invariants for pseudocellular embeddings and discuss connections to other polynomial invariants.
