Flexible circuits in the d-dimensional rigidity matroid

Grasegger, Georg and Guler, Hakan and Jackson, Bill and Nixon, Anthony (2022) Flexible circuits in the d-dimensional rigidity matroid. Journal of Graph Theory, 100 (2). pp. 315-360. ISSN 0364-9024

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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.