  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.
