Characterizing generic global rigidity
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Publication:3584599
DOI10.1353/AJM.0.0132zbMATH Open1202.52020arXiv0710.0926OpenAlexW2107850743MaRDI QIDQ3584599
Author name not available (Why is that?)
Publication date: 30 August 2010
Published in: (Search for Journal in Brave)
Abstract: A d-dimensional framework is a graph and a map from its vertices to E^d. Such a framework is globally rigid if it is the only framework in E^d with the same graph and edge lengths, up to rigid motions. For which underlying graphs is a generic framework globally rigid? We answer this question by proving a conjecture by Connelly, that his sufficient condition is also necessary: a generic framework is globally rigid if and only if it has a stress matrix with kernel of dimension d+1, the minimum possible. An alternate version of the condition comes from considering the geometry of the length-squared mapping l: the graph is generically locally rigid iff the rank of l is maximal, and it is generically globally rigid iff the rank of the Gauss map on the image of l is maximal. We also show that this condition is efficiently checkable with a randomized algorithm, and prove that if a graph is not generically globally rigid then it is flexible one dimension higher.
Full work available at URL: https://arxiv.org/abs/0710.0926
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