Structural characterizations and Obstructions to Planar-Rips complexes

Given a scale parameter r>0, the Vietoris-Rips complex of X R² (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.