# 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}
}
• Published 4 March 2012
• Mathematics
• arXiv: Geometric Topology
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.
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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
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SIAM J. Discret. Math.
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• 2014
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• Materials Science
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A Characterization of Generically Rigid Frameworks on Surfaces of Revolution
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SIAM J. Discret. Math.
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## References

SHOWING 1-10 OF 52 REFERENCES
Frameworks with Forced Symmetry I: Reflections and Rotations
• Mathematics
Discret. 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 frameworks with orientation-preserving crystallographic symmetry
• Mathematics
• 2011
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
• 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
Frameworks with forced symmetry II: orientation-preserving crystallographic groups
• Mathematics
• 2013
We give a combinatorial characterization of minimally rigid planar frameworks with orientation-preserving crystallographic symmetry, under the constraint of forced symmetry. The main theorems are
On graphs and rigidity of plane skeletal structures
• G. Laman
• Mathematics, Materials Science
• 1970
SummaryIn this paper the combinatorial properties of rigid plane skeletal structures are investigated. Those properties are found to be adequately described by a class of graphs.
Rigid components in fixed-lattice and cone frameworks
• Mathematics, Computer Science
CCCG
• 2011
If the order of rotation is part of the input, this work gives an O(n^4) algorithm for deciding rigidity; in the case where the rotation's order is 3, a more specialized algorithm solves all the fundamental algorithmic rigidity problems in O( n^2) time.
Minimally rigid periodic graphs
• Mathematics
• 2011
We prove a rigidity theorem of Maxwell–Laman type for periodic frameworks in arbitrary dimension.
Rigidity of Frameworks Supported on Surfaces
• Mathematics
SIAM J. Discret. Math.
• 2012
A theorem of Laman gives a combinatorial characterisation of the graphs that admit a realisation as a minimally rigid generic bar-joint framework in $\bR^2$ whose vertices are constrained to move on a two-dimensional smooth submanifold $\M$.
A Note on [k, l]-sparse Graphs
• Mathematics
• 2006
In this note we provide a Henneberg-type constructive characterization theorem of [k, l]-sparse graphs, that is, the graphs for which the number of induced edges in any subset X of nodes is at most