标题： 结构表征和限制平面-Rips复形的因素 显示英文标题

标题： Structural characterizations and Obstructions to Planar-Rips complexes

摘要： 给定一个尺度参数$r>0$，在通常的欧几里得度量下，$X \subset \mathbb{R}^2$的 Vietoris-Rips 复形（称为平面-Rips 复形）是一个有限的单纯复形，其单形是$X$的子集，直径不超过$r$。 本文重点在于表征可以或不能作为平面-Rips 复形实现的单纯复形。 在 Adamszek 等人先前工作的基础上，我们按单纯同构分类了所有可接受平面-Rips 结构的$n$维伪流形和弱伪流形，并进一步表征了二维、纯且闭合的平面-Rips 复形。 此外，已经引入了平面-Rips 复形的障碍概念，为识别禁止的平面-Rips 结构的算法方法奠定了基础。 我们还探讨了平面-Rips 复形与称为圆盘图的交图类之间的相关性，建立了这两个类之间的自然同构。 同时，我们的关于平面-Rips 复形的发现已被整理成单位圆盘图的形式，供有兴趣的读者参考，描绘了拓扑和代数方法的重要性。 已经推导出了一些平面-Rips 复形的结构和几何性质，这些性质具有独立的兴趣。 显示英文摘要

摘要： Given a scale parameter $r>0$, the Vietoris-Rips complex of $X \subset \mathbb{R}^2$ (referred to as planar-Rips complex) under the usual Euclidean metric is a (finite) simplicial complex whose simplices are subsets of $X$ with diameter at most $r$. This paper focuses on characterizing the simplicial complexes that can or cannot be realized as planar-Rips complexes. Building on the prior work of Adamszek et al., we classify, up to simplicial isomorphism, all $n$-dimensional pseudomanifolds and weak-pseudomanifolds that admit a planar-Rips structure, and further characterize two-dimensional, pure, and closed planar-Rips complexes. Additionally, the notion of obstructions to planar-Rips complexes has been introduced, laying the groundwork for algorithmic approaches to identifying forbidden planar-Rips structures. We also explore the correlations between planar-Rips complexes and the class of intersection graphs called disk graphs, establishing a natural isomorphism between the two classes. Parallelly, our findings on planar-Rips complexes have been consolidated in terms of unit disk graphs for interested readers, depicting the significance of the topological and algebraic approaches. Several structural and geometric properties of planar-Rips complexes have been derived that are of independent interest.