Frameworks with Forced Symmetry I: Reflections and Rotations

@article{Malestein2013FrameworksWF,
  title={Frameworks with Forced Symmetry I: Reflections and Rotations},
  author={Justin Malestein and Louis Theran},
  journal={Discrete \& Computational Geometry},
  year={2013},
  volume={54},
  pages={339-367}
}
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 reflection. To establish these results, we develop a new technique for deriving linear representations of sparsity matroids on colored graphs and extend the direction network method of proving rigidity characterizations to handle reflections. 

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References

SHOWING 1-10 OF 40 REFERENCES

Generic Rigidity with Forced Symmetry and Sparse Colored Graphs

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

Frameworks with forced symmetry II: orientation-preserving crystallographic groups

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

Generic rigidity of reflection frameworks

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

Generic rigidity of frameworks with orientation-preserving crystallographic symmetry

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 combinatorial rigidity of periodic frameworks

Gain-Sparsity and Symmetry-Forced Rigidity in the Plane

It is shown that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroid defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group.

Periodic Rigidity on a Variable Torus Using Inductive Constructions

This paper proves a recursive characterisation of generic rigidity for frameworks periodic with respect to a partially variable lattice with variants of the Henneberg operations used frequently in rigidity theory.

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.

Voltage-Graphic Matroids

From an integer valued function f we obtain, in a natural way, a matroid Mf on the domain of f. We show that the class M of matroids so obtained is closed under restriction, contraction, duality,

Construction of D-Graphs Related to Periodic Tilings

An algorithm is presented which allows to derive classification methods concerning periodic tilings in any dimension, theoretically, and which yields the complete enumeration of non-isomorphic three-dimensional D−graphs with 5 elements.