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Flexible circuits in the d ‐dimensional rigidity matroid
Author(s) -
Grasegger Georg,
Guler Hakan,
Jackson Bill,
Nixon Anthony
Publication year - 2022
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.22780
Subject(s) - matroid , mathematics , combinatorics , rigidity (electromagnetism) , electronic circuit , rank (graph theory) , graph , discrete mathematics , topology (electrical circuits) , electrical engineering , structural engineering , engineering
A bar‐joint framework ( G , p ) in R d is rigid if the only edge‐length preserving continuous motions of the vertices arise from isometries of R d . It is known that, when ( G , p ) is generic, its rigidity depends only on the underlying graph G , and is determined by the rank of the edge set of G in the generic d ‐dimensional rigidity matroid ℛ d . Complete combinatorial descriptions of the rank function of this matroid are known when d = 1 , 2 , and imply that all circuits in ℛ d are generically rigid in R d when d = 1 , 2 . Determining the rank function of ℛ d is a long standing open problem when d ≥ 3 , and the existence of nonrigid circuits in ℛ d for d ≥ 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 ℛ d which have at most d + 6 vertices.