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Gain-sparsity and symmetry-forced rigidity in the plane

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<mark>Journal publication date</mark>03/2016
<mark>Journal</mark>Discrete and Computational Geometry
Issue number2
Volume55
Number of pages59
Pages (from-to)314-372
Publication StatusPublished
Early online date1/02/16
<mark>Original language</mark>English

Abstract

We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2k with odd k, unifying and extending previous work on this subject. We also explore the matroidal background of our results and show that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroids defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group. The proofs are based on new Henneberg type inductive constructions of the gain graphs that correspond to the bases of the matroids in question, which can also be seen as symmetry preserving graph operations in the original graph.