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Generically Globally Rigid Graphs Have Generic Universally Rigid Frameworks

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Abstract

We show that any graph that is generically globally rigid in ℝd has a realization in ℝd that is both generic 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.

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Robert Connelly

  2. School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, Scotland, UK

    Louis Theran

  3. School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138, USA

    Steven J. Gortler

Authors
  1. Robert Connelly
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  2. Steven J. Gortler
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  3. Louis Theran
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Additional information

An earlier version of this paper [15] had the title “Generic global and universal rigidity”.

Partially supported by NSF grant DMS-1564493.

Partially supported by NSF grant DMS-1564473.

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Connelly, R., Gortler, S.J. & Theran, L. Generically Globally Rigid Graphs Have Generic Universally Rigid Frameworks. Combinatorica 40, 1–37 (2020). https://doi.org/10.1007/s00493-018-3694-4

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  • DOI: https://doi.org/10.1007/s00493-018-3694-4

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