Generically globally rigid graphs have generic universally rigid frameworks

Authors: Robert Connelly, Steven J. Gortler, and Louis Theran
Journal: Combinatorica, 2020.
Full text: arXiv DOI

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 Lovász, 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.