Gain-Sparsity and Symmetry-Forced Rigidity in the Plane

Tibor Jordán, Viktória E. Kaszanitzky, Shin ichi Tanigawa

Research output: Contribution to journalArticle

6 Citations (Scopus)


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 and Computational Geometry
Issue number2
Publication statusPublished - Mar 1 2016



  • Frame matroids
  • Frameworks
  • Group-labeled graphs
  • Infinitesimal rigidity
  • Rigidity matroids
  • Rigidity of graphs
  • Symmetry

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Cite this