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

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.