# Generic Rigidity with Forced Symmetry and Sparse Colored Graphs

@article{Malestein2014GenericRW, title={Generic Rigidity with Forced Symmetry and Sparse Colored Graphs}, author={Justin Malestein and Louis Theran}, journal={arXiv: Geometric Topology}, year={2014}, pages={227-252} }

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.

## 17 Citations

Frameworks with Forced Symmetry I: Reflections and Rotations

- MathematicsDiscret. Comput. Geom.
- 2015

We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a…

Generic rigidity of reflection frameworks

- Mathematics
- 2012

We give a combinatorial characterization of generic minimally rigid reflection frameworks. The main new idea is to study a pair of direction networks on the same graph such that one admits faithful…

Henneberg constructions and covers of cone-Laman graphs

- Mathematics
- 2012

We give Henneberg-type constructions for three families of sparse colored graphs arising in the rigidity theory of periodic and other forced symmetric frameworks. The proof method, which works with…

Infinitesimal Rigidity of Symmetric Bar-Joint Frameworks

- MathematicsSIAM J. Discret. Math.
- 2015

Using these new tools, combinatorial characterizations of infinitesimally rigid two-dimensional bar-joint frameworks whose joints are positioned as generically as possible subject to the symmetry constraints imposed by a reflection, a half-turn, or a threefold rotation in the plane are established.

Title Infinitesimal Rigidity of Symmetric Bar-Joint Frameworks

- Mathematics
- 2018

We propose new symmetry-adapted rigidity matrices to analyze the infinitesimal rigidity of bar-joint frameworks of arbitrary-dimension with Abelian point group symmetries. These matrices define new…

Ultrarigid periodic frameworks

- MathematicsArXiv
- 2014

We give an algebraic characterization of when a $d$-dimensional periodic framework has no non-trivial, symmetry preserving, motion for any choice of periodicity lattice. Our condition is decidable,…

Symmetry-forced rigidity of frameworks on surfaces

- Mathematics
- 2013

A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been…

Symmetric and Spectral Realizations of Highly Symmetric Graphs

- Mathematics
- 2020

A realization of a graph $G=(V,E)$ is a map $v\colon V\to\Bbb R^d$ that assigns to each vertex a point in $d$-dimensional Euclidean space. We study graph realizations from the perspective of…

62 RIGIDITY OF SYMMETRIC FRAMEWORKS

- Materials Science
- 2016

Since symmetry is ubiquitous in both man-made structures (e.g., buildings or mechanical linkages) and in structures found in nature (e.g., proteins or crystals), it is natural to consider the impact…

A Characterization of Generically Rigid Frameworks on Surfaces of Revolution

- MathematicsSIAM J. Discret. Math.
- 2014

A foundational theorem of Laman provides a counting characterization of the finite simple graphs whose generic bar-joint frameworks in two dimensions are infinitesimally rigid, which is a Laman-type theorem for frameworks on algebraic surfaces with a 1-dimensional space of tangential motions.

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We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a…

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We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial…

Generic rigidity of reflection frameworks

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We give a combinatorial characterization of generic minimally rigid reflection frameworks. The main new idea is to study a pair of direction networks on the same graph such that one admits faithful…

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