具有强自对偶性的几何实现 第一部分：射影平面的非二分四边形剖分与强对合自对偶图

中文摘要 AI 辅助

在本系列文章中，我们研究两类图及其各种几何表示。第一类由强对合自对偶图（SISD图）组成；第二类由射影平面的非二分四边形剖分（NBQP图）组成。这些类自然出现在许多几何问题中。例如，NBQP图与$\mathbb{R}^3$中的直径图、成对相交圆和伪圆的族相切图、广义thrackles以及负自极多胞体的主对角线形成的图相关联。另一方面，SISD图对应于负自极多胞体的骨架、自对偶锥和极值球多胞体。在第一部分中，我们展示了射影平面的非二分四边形剖分与2-连通强对合自对偶映射之间的自然一一对应关系。特别地，我们证明了2-连通强对合自对偶映射的约化顶点-面关联结构是射影平面的非二分四边形剖分，并且射影平面的每个非二分四边形剖分都以这种方式产生。这使我们能够将自然性质从一类转换到另一类。这些结果为系列的后续部分奠定了基础。

英文摘要

In this series of articles, we study two graph classes and their various geometric representations. The first class consists of strongly involutive self-dual graphs (SISD graphs); the second consists of non-bipartite quadrangulations of the projective plane (NBQP graphs). These classes naturally arise in many geometric problems. For example, NBQP graphs are connected to diameter graphs in $\mathbb{R}^3$, tangency graphs of families of pairwise intersecting circles and pseudocircles, generalized thrackles, and graphs formed by the main diagonals of negatively self-polar polytopes. On the other hand, SISD graphs correspond to the skeletons of negatively self-polar polytopes, self-dual cones, and extremal ball-polytopes. In Part I we show a natural one-to-one correspondence between non-bipartite quadrangulations of the projective plane and 2-connected strongly involutive self-dual maps. In particular, we show that the reduced vertex-face incidence structure of a 2-connected strongly involutive self-dual map is a non-bipartite quadrangulation of the projective plane, and every non-bipartite quadrangulation of the projective plane arises that way. This allows us to translate natural properties from one class to the other. These results lay the foundation for later parts of the series.

发表机构

• HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究网络阿尔弗雷德·雷尼数学研究所）
• ELTE Eötvös Loránd University（厄特沃什·罗兰大学）

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25 pages, 24 figures