Measurement Isomorphism of Graphs

TL;DR

This paper introduces d-measurement isomorphism for graphs, equating it with Whitney's 2-isomorphism, and explores the relationship between these graph properties.

Contribution

It defines d-measurement isomorphism and establishes its equivalence to Whitney's 2-isomorphism for graphs.

Findings

d-measurement sets characterize graph embeddings

d-measurement isomorphism coincides with Whitney's 2-isomorphism

provides a new perspective on graph isomorphism

Abstract

The d-measurement set of a graph is its set of possible squared edge lengths over all d-dimensional embeddings. In this note, we define a new notion of graph isomorphism called d-measurement isomorphism. Two graphs are d-measurement isomorphic if there is agreement in their d-measurement sets. A natural question to ask is "what can be said about two graphs that are d-measurement isomorphic?" In this note, we show that this property coincides with the 2-isomorphism property studied by Whitney.