Gain-Sparsity and Symmetry-Forced Rigidity in the Plane

T. Jordán, Viktória E. Kaszanitzky, Shin ichi Tanigawa

Research output: Contribution to journalArticle

4 Citations (Scopus)

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 languageEnglish
Pages (from-to)314-372
Number of pages59
JournalDiscrete & Computational Geometry
Volume55
Issue number2
DOIs
Publication statusPublished - Mar 1 2016

Fingerprint

Matroid
Sparsity
Rigidity
Symmetry Group
Symmetry
Order of symmetry
Graph in graph theory
Quotient Graph
Rotational symmetry
Point groups
Crystal symmetry
Isomorphic
Odd
Orbit
Orbits
Framework

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

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

In: Discrete & Computational Geometry, Vol. 55, No. 2, 01.03.2016, p. 314-372.

Research output: Contribution to journalArticle

Jordán, T. ; Kaszanitzky, Viktória E. ; Tanigawa, Shin ichi. / Gain-Sparsity and Symmetry-Forced Rigidity in the Plane. In: Discrete & Computational Geometry. 2016 ; Vol. 55, No. 2. pp. 314-372.
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