Edge‐Length Preserving Embeddings of Graphs Between Normed Spaces

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 is said to be ‐embeddable if any set of induced edge lengths from an embedding of into a normed space can also be realised by an embedding of into a normed space . This property, being minor‐closed, can be characterized by a finite list of forbidden minors. Following the establishment of fundamental results about ‐embeddability, we identify sufficient conditions under which it implies independence with respect to the associated rigidity matroids for and . We show that the spaces and serve as two natural extreme spaces of embeddability and discuss ‐embeddability for varying . We provide a complete characterization of ‐embeddable graphs for the specific case when is 2‐dimensional and is infinite‐dimensional.