Characterizing Generic Global Rigidity

TL;DR

This paper characterizes when a generic framework in Euclidean space is globally rigid, proving a conjecture that links global rigidity to the existence of a stress matrix with minimal kernel dimension, and providing efficient verification methods.

Contribution

It proves Connelly's conjecture that a generic framework is globally rigid iff it has a stress matrix with kernel of dimension d+1, and introduces an efficient randomized algorithm for checking this condition.

Findings

A generic framework is globally rigid iff it has a stress matrix with kernel of dimension d+1.

The condition for global rigidity can be checked efficiently using a randomized algorithm.

Non-globally rigid graphs are flexible in one higher dimension.

Abstract

A d-dimensional framework is a graph and a map from its vertices to E^d. Such a framework is globally rigid if it is the only framework in E^d with the same graph and edge lengths, up to rigid motions. For which underlying graphs is a generic framework globally rigid? We answer this question by proving a conjecture by Connelly, that his sufficient condition is also necessary: a generic framework is globally rigid if and only if it has a stress matrix with kernel of dimension d+1, the minimum possible. An alternate version of the condition comes from considering the geometry of the length-squared mapping l: the graph is generically locally rigid iff the rank of l is maximal, and it is generically globally rigid iff the rank of the Gauss map on the image of l is maximal. We also show that this condition is efficiently checkable with a randomized algorithm, and prove that if a graph is not generically globally rigid then it is flexible one dimension higher.