Generically globally rigid graphs have generic universally rigid frameworks

Robert Connelly; Steven Gortler; Louis Simon Theran

doi:10.1007/s00493-018-3694-4

Generically globally rigid graphs have generic universally rigid frameworks

Robert Connelly, Steven Gortler, Louis Simon Theran

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show that any graph that is generically globally rigid in ℝ^d has a realization in ℝ^d both generic and universally rigid. It also must have a realization in ℝ^d that is both infinitesimally rigid 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.

Original language	English
Pages (from-to)	1-37
Journal	Combinatorica
Volume	40
DOIs	https://doi.org/10.1007/s00493-018-3694-4
Publication status	Published - 22 Mar 2020

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Copyright © 2020 János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1007/s00493-018-3694-4.
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http://arxiv.org/abs/1604.07475

Louis Simon Theran
- louis.theranst-andrews.acuk
- Pure Mathematics - Lecturer
- Centre for Interdisciplinary Research in Computational Algebra
Person: Academic

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Generically globally rigid graphs have generic universally rigid frameworks

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