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arXiv · 1604.07475

Generically globally rigid graphs have generic universally rigid frameworks

Robert Connelly·Steven J. Gortler·Louis Theran
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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.

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Robert Connelly, Steven J. Gortler, Louis Theran. 2016-04-25. Generically globally rigid graphs have generic universally rigid frameworks. https://arxiv.org/abs/1604.07475

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