Generically Globally Rigid Graphs Have Generic Universally Rigid Frameworks


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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Corresponding author

Correspondence to Louis Theran.

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).

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Mathematics Subject Classification (2010)

  • 52C25
  • 05C62