Generically Globally Rigid Graphs Have Generic Universally Rigid Frameworks

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

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