Classifying the globally rigid edge-transitive graphs and distance-regular graphs in the plane

TL;DR

This paper extends the classification of globally rigid graphs in the plane to include all graphs determined by their automorphism group, focusing on edge-transitive and distance-regular graphs using degrees.

Contribution

It provides a complete characterization of which edge-transitive and distance-regular graphs are globally rigid based on their degrees.

Findings

Edge-transitive graphs are globally rigid if their degrees meet specific criteria.

Distance-regular graphs' global rigidity is characterized by their minimal and maximal degrees.

Extension of prior vertex-transitive graph results to broader classes of graphs.

Abstract

A graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge-length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius characterised exactly which vertex-transitive graphs are globally rigid solely by their degree and maximal clique number, two easily computable parameters for vertex-transitive graphs. In this short note we will extend this characterisation to all graphs that are determined by their automorphism group. We do this by characterising exactly which edge-transitive graphs and distance-regular graphs are globally rigid by their minimal and maximal degrees.