Generically globally rigid graphs have generic universally rigid frameworks

Abstract

We show that any graph that is generically globally rigid in ℝd has a realization in ℝd both generic and universally rigid. It also must have a realization in ℝd that is both infinitesimally rigid and universally rigid. This also implies that the graph also must have a realization in ℝd 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.