Generically globally rigid graphs have generic universally rigid frameworks

TL;DR

This paper proves that any generically globally rigid graph in Euclidean space has a realization that is both generic and universally rigid, providing a certificate for global rigidity through a constructive approach.

Contribution

It establishes the existence of a generic, universally rigid realization for any generically globally rigid graph, linking combinatorial rigidity with geometric realizations.

Findings

Existence of a generic, universally rigid realization for such graphs

Use of Lovász, Saks, and Schrijver's algorithm to construct orthogonal representations

Application of Alfakih's result to derive stress matrices and universal rigidity

Abstract

We show that any graph that is generically globally rigid in $\mathbb{R}^d$ has a realization in $\mathbb{R}^d$ that is both generic and universally rigid. This also implies that the graph also must have a realization in $\mathbb{R}^d$ that is both infinitesimally rigid and universally rigid; such a realization serves as a certificate of generic global rigidity. Our approach involves an algorithm by Lov\'asz, Saks and Schrijver that, for a sufficiently connected graph, constructs a general position orthogonal representation of the vertices, and a result of Alfakih that shows how this representation leads to a stress matrix and a universally rigid framework of the graph.