Gain-sparsity and symmetry-forced rigidity in the plane

Jordán, Tibor and Kaszanitzky, Viktoria Eszter and Tanigawa, Shin-ichi (2016) Gain-sparsity and symmetry-forced rigidity in the plane. Discrete and Computational Geometry, 55 (2). pp. 314-372. ISSN 0179-5376

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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.

Item Type:
Journal Article
Journal or Publication Title:
Discrete and Computational Geometry
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600/2614
Subjects:
ID Code:
80996
Deposited By:
Deposited On:
17 Aug 2016 12:10
Refereed?:
Yes
Published?:
Published
Last Modified:
25 Feb 2020 07:39