Structural characterizations and Obstructions to Planar-Rips complexes
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.01082
- arXiv:
- arXiv:2406.01082
- Bibcode:
- 2024arXiv240601082S
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Algebraic Topology;
- Mathematics - Geometric Topology;
- Mathematics - Metric Geometry;
- 05E45;
- 68U05;
- 05C10