Flexible circuits in the d‐dimensional rigidity matroid - Grasegger, Georg; Guler, Hakan; Jackson, Bill; Nixon, Anthony (Wiley, 2021)
AbstractA bar‐joint framework in is rigid if the only edge‐length preserving continuous motions of the vertices arise from isometries of . It is known that, when is generic, its rigidity depends only on the underlying graph , and is determined by the rank of the edge set of in the generic ‐dimensional rigidity matroid . Complete combinatorial descriptions of the rank function of this matroid are known when , and imply that all circuits in are generically rigid in when . Determining the rank function of is a long standing open problem when , and the existence of nonrigid circuits in for 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 which have at most vertices.
Keywords: rigidity matroid; Metric Geometry (math.MG); Combinatorial aspects of matroids and geometric lattices; rigid graph; C25, 05C10; Planar graphs; geometric and topological aspects of graph theory; Mathematics - Metric Geometry; Rigidity and flexibility of structures (aspects of discrete geometry); FOS: Mathematics; bar-joint framework; Mathematics - Combinatorics; Combinatorics (math.CO); flexible circuit