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

TL;DR

This paper classifies all n-dimensional pseudomanifolds and weak-pseudomanifolds that can be realized as Vietoris-Rips complexes of planar point sets, and explores their properties and related unit disk graphs.

Contribution

It provides a comprehensive classification of planar-Rips complexes and their homotopy types, introducing obstructions and structural properties, advancing understanding of their geometric and combinatorial nature.

Findings

Classified all n-dimensional pseudomanifolds as Vietoris-Rips complexes of planar points.

Identified structural properties and obstructions in planar-Rips complexes.

Described a class of unit disk graphs with uniform maximal clique sizes.

Abstract

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.