标题： 平面-Rips复形及其对应的单位disk图的分类 显示英文标题

标题： On the Classification of Planar-Rips complexes and their corresponding unit disk graphs

摘要： 给定一个度量空间$(X,d)$，在尺度为$r >0$的情况下，$X$的 Vietoris-Rips 复形是一个单纯复形，其单形是$X$中所有直径小于$r$的有限子集。 在本文中，我们根据单纯同构对所有可以作为平面点集的 Vietoris-Rips 复形的$n$维伪流形和弱伪流形进行分类。 我们进一步根据同伦对二维、纯且闭合的平面-Rips 复形进行分类。 此外，我们探讨了继承性质，并在平面-Rips 复形中引入了障碍的概念。 我们还综合了研究成果，描述了一类单位 disk 图，其所有极大团的基数相同。 平面-Rips 复形的若干结构和几何性质也已被推导出来。 显示英文摘要

摘要： Given a metric space $(X,d)$, the Vietoris-Rips complex of $X$ at a scale of $r >0$ is a simplicial complex whose simplices are all those finite subsets of $X$ with diameter less than $r$. In this paper, we classify, up to simplicial isomorphism, all $n$-dimensional pseudomanifolds and weak-pseudomanifolds that can be realized as a Vietoris-Rips complex of planar point sets. We further classify two-dimensional, pure, and closed planar-Rips complexes up to homotopy. Additionally, we explore the hereditary properties and introduce the notion of obstructions in planar-Rips complexes. We also consolidate our findings to describe a class of unit disk graphs, having all maximal cliques of same cardinality. Several structural and geometric properties of planar-Rips complexes have also been derived.