TR-2012-17

Gain-sparsity and Symmetry-forced Rigidity in the Plane

Tibor Jordán, Viktória Kaszanitzky, Shin-ichi Tanigawa

Published in:
Discrete Comput Geom 55, 314–372 (2016). DOI link

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 cyclic or odd-order dihedral symmetry, 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.

Bibtex entry:

@techreport{egres-12-17,

AUTHOR	=	{Jord{\'a}n, Tibor and Kaszanitzky, Vikt{\'o}ria and Tanigawa, Shin-ichi},
TITLE	=	{Gain-sparsity and Symmetry-forced Rigidity in the Plane},
NOTE	=	{{\tt egres.elte.hu}},
INSTITUTION	=	{Egerv{\'a}ry Research Group, Budapest},
YEAR	=	{2012},
NUMBER	=	{TR-2012-17}

}