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Rights statement: This is the peer reviewed version of the following article: Grasegger, G., Guler, H., Jackson, B., and Nixon, A., Flexible circuits in the d-dimensional rigidity matroid, J. Graph. Theory. 2022; 100: 315– 330. https://doi.org/10.1002/jgt.22780 which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1002/jgt.22780 This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving.

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Flexible circuits in the d-dimensional rigidity matroid

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Journal publication date 30/06/2022 Journal of Graph Theory 2 100 16 315-360 Published 6/12/21 English

Abstract

A bar-joint framework (퐺,푝) in ℝ푑 is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of ℝ푑 . It is known that, when (퐺,푝) is generic, its rigidity depends only on the underlying graph 퐺 , and is determined by the rank of the edge set of 퐺 in the generic 푑 -dimensional rigidity matroid ℛ푑 . Complete combinatorial descriptions of the rank function of this matroid are known when 푑=1,2 , and imply that all circuits in ℛ푑 are generically rigid in ℝ푑 when 푑=1,2 . Determining the rank function of ℛ푑 is a long standing open problem when 푑≥3 , and the existence of nonrigid circuits in ℛ푑 for 푑≥3 is a major contributing factor to why this problem is so difficult. We begin a study of nonrigid circuits by characterising the nonrigid circuits in ℛ푑 which have at most 푑+6 vertices.

Bibliographic note

This is the peer reviewed version of the following article: Grasegger, G., Guler, H., Jackson, B., and Nixon, A., Flexible circuits in the d-dimensional rigidity matroid, J. Graph. Theory. 2022; 100: 315– 330. https://doi.org/10.1002/jgt.22780 which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1002/jgt.22780 This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving.