Edge‐Length Preserving Embeddings of Graphs Between Normed Spaces

Dewar, Sean and Kastis, Eleftherios and Kitson, Derek and Sims, William (2026) Edge‐Length Preserving Embeddings of Graphs Between Normed Spaces. Journal of Graph Theory. ISSN 0364-9024

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Abstract

The concept of graph embeddability, 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 ) $G=(V,E)$ is said to be ( X , Y ) $(X,Y)$ ‐embeddable if any set of induced edge lengths from an embedding of G $G$ into a normed space Y $Y$ can also be realised by an embedding of G $G$ into a normed space X $X$ . This property, being minor‐closed, can be characterized by a finite list of forbidden minors. Following the establishment of fundamental results about ( X , Y ) $(X,Y)$ ‐embeddability, we identify sufficient conditions under which it implies independence with respect to the associated rigidity matroids for X $X$ and Y $Y$ . We show that the spaces ℓ 2 ${\ell }_{2}$ and ℓ ∞ ${\ell }_{\infty }$ serve as two natural extreme spaces of embeddability and discuss ( X , ℓ p ) $(X,{\ell }_{p})$ ‐embeddability for varying p $p$ . We provide a complete characterization of ( X , Y ) $(X,Y)$ ‐embeddable graphs for the specific case when X $X$ is 2‐dimensional and Y $Y$ is infinite‐dimensional.