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Gain-sparsity and symmetry-forced rigidity in the plane

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Gain-sparsity and symmetry-forced rigidity in the plane. / Jordán, Tibor ; Kaszanitzky, Viktoria Eszter; Tanigawa, Shin-ichi.
In: Discrete and Computational Geometry, Vol. 55, No. 2, 03.2016, p. 314-372.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Jordán, T, Kaszanitzky, VE & Tanigawa, S 2016, 'Gain-sparsity and symmetry-forced rigidity in the plane', Discrete and Computational Geometry, vol. 55, no. 2, pp. 314-372. https://doi.org/10.1007/s00454-015-9755-1

APA

Jordán, T., Kaszanitzky, V. E., & Tanigawa, S. (2016). Gain-sparsity and symmetry-forced rigidity in the plane. Discrete and Computational Geometry, 55(2), 314-372. https://doi.org/10.1007/s00454-015-9755-1

Vancouver

Jordán T, Kaszanitzky VE, Tanigawa S. Gain-sparsity and symmetry-forced rigidity in the plane. Discrete and Computational Geometry. 2016 Mar;55(2):314-372. Epub 2016 Feb 1. doi: 10.1007/s00454-015-9755-1

Author

Jordán, Tibor ; Kaszanitzky, Viktoria Eszter ; Tanigawa, Shin-ichi. / Gain-sparsity and symmetry-forced rigidity in the plane. In: Discrete and Computational Geometry. 2016 ; Vol. 55, No. 2. pp. 314-372.

Bibtex

@article{83761ac5995e47fca1f42b185f91df0d,
title = "Gain-sparsity and symmetry-forced rigidity in the plane",
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.",
keywords = "Infinitesimal rigidity, Frameworks, Symmetry, Rigidity of graphs, Rigidity matroids, Group-labeled graphs, Fame matroids",
author = "Tibor Jord{\'a}n and Kaszanitzky, {Viktoria Eszter} and Shin-ichi Tanigawa",
year = "2016",
month = mar,
doi = "10.1007/s00454-015-9755-1",
language = "English",
volume = "55",
pages = "314--372",
journal = "Discrete and Computational Geometry",
issn = "0179-5376",
publisher = "Springer New York",
number = "2",

}

RIS

TY - JOUR

T1 - Gain-sparsity and symmetry-forced rigidity in the plane

AU - Jordán, Tibor

AU - Kaszanitzky, Viktoria Eszter

AU - Tanigawa, Shin-ichi

PY - 2016/3

Y1 - 2016/3

N2 - 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.

AB - 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.

KW - Infinitesimal rigidity

KW - Frameworks

KW - Symmetry

KW - Rigidity of graphs

KW - Rigidity matroids

KW - Group-labeled graphs

KW - Fame matroids

U2 - 10.1007/s00454-015-9755-1

DO - 10.1007/s00454-015-9755-1

M3 - Journal article

VL - 55

SP - 314

EP - 372

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 2

ER -