Edge-length preserving embeddings of graphs between normed spaces

TL;DR

This paper investigates conditions under which graphs can be embedded into different normed spaces while preserving edge lengths, extending the concept of graph flattenability and characterizing such embeddings for specific space pairs.

Contribution

It introduces new sufficient conditions linking flattenability to rigidity matroids and provides a complete characterization for 2D to infinite-dimensional space embeddings.

Findings

Spaces $\, ext{ell}_2$ and $ ext{ell}_ty$ are natural extremes for flattenability.

Characterization of $(X,Y)$-flattenable graphs when $X$ is 2D and $Y$ is infinite-dimensional.

Identifies finite forbidden minors for flattenability property.

Abstract

The concept of graph flattenability, initially formalized by Belk and Connelly and later expanded by Sitharam and Willoughby, extends the question of embedding finite metric spaces into a given normed space. A finite simple graph $G=(V,E)$ is said to be $(X,Y)$-flattenable if any set of induced edge lengths from an embedding of $G$ into a normed space $Y$ can also be realised by an embedding of $G$ into a normed space $X$. This property, being minor-closed, can be characterized by a finite list of forbidden minors. Following the establishment of fundamental results about $(X,Y)$-flattenability, we identify sufficient conditions under which it implies independence with respect to the associated rigidity matroids for $X$ and $Y$. We show that the spaces $\ell_2$ and $\ell_\infty$ serve as two natural extreme spaces of flattenability and discuss $(X, \ell_p )$-flattenability for varying $p$. We provide a complete characterization of $(X,Y)$-flattenable graphs for the specific case when $X$ is 2-dimensional and $Y$ is infinite-dimensional.