Connecting Global and Universal Rigidity

Abstract: A d-dimensional framework is an embedding of the vertices and edges of a graph in Euclidean space. A d-dimensional framework is globally rigid if every other d-dimensional framework with the same edge lengths has the same pairwise distances between the vertices. A graph is generically globally rigid in dimension d (d-GGR) if every generic framework is globally rigid. The d-dimensional framework of a d-GGR graph is universally rigid if for d' greater than or equal to d every d'-dimensional framework with the same edge lengths has the same pairwise distances between the vertices. We establish a strong connection between global and universal rigidity by showing that all 1 and 2-GGR graphs and an infinite number of higher dimensional d-GGR graphs have a generic universally rigid framework.