### 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.

Original language | English |
---|---|

Pages (from-to) | 314-372 |

Number of pages | 59 |

Journal | Discrete & Computational Geometry |

Volume | 55 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 1 2016 |

### Fingerprint

### Keywords

- Frame matroids
- Frameworks
- Group-labeled graphs
- Infinitesimal rigidity
- Rigidity matroids
- Rigidity of graphs
- Symmetry

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Discrete & Computational Geometry*,

*55*(2), 314-372. https://doi.org/10.1007/s00454-015-9755-1

**Gain-Sparsity and Symmetry-Forced Rigidity in the Plane.** / Jordán, T.; Kaszanitzky, Viktória E.; Tanigawa, Shin ichi.

Research output: Contribution to journal › Article

*Discrete & Computational Geometry*, vol. 55, no. 2, pp. 314-372. https://doi.org/10.1007/s00454-015-9755-1

}

TY - JOUR

T1 - Gain-Sparsity and Symmetry-Forced Rigidity in the Plane

AU - Jordán, T.

AU - Kaszanitzky, Viktória E.

AU - Tanigawa, Shin ichi

PY - 2016/3/1

Y1 - 2016/3/1

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 - Frame matroids

KW - Frameworks

KW - Group-labeled graphs

KW - Infinitesimal rigidity

KW - Rigidity matroids

KW - Rigidity of graphs

KW - Symmetry

UR - http://www.scopus.com/inward/record.url?scp=84957990668&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957990668&partnerID=8YFLogxK

U2 - 10.1007/s00454-015-9755-1

DO - 10.1007/s00454-015-9755-1

M3 - Article

VL - 55

SP - 314

EP - 372

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 2

ER -