Gain-Sparsity and Symmetry-Forced Rigidity in the Plane

• Published in: Discrete & computational geometry Vol. 55; no. 2; pp. 314 - 372
• Main Authors: Jordán, Tibor, Kaszanitzky, Viktória E., Tanigawa, Shin-ichi
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2016; Springer Nature B.V
• Subjects: Combinatorial analysis; Combinatorics; Computational geometry; Computational Mathematics and Numerical Analysis; Gain; Geometry; Graph theory; Graphs; Mathematics; Mathematics and Statistics; Planes; Rigidity; Symmetry; Frame matroids; Infinitesimal rigidity; Frameworks; Primary 52C25; Group-labeled graphs; Secondary 05B35; 68R10; Rigidity matroids; Symmetry; Rigidity of graphs
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-015-9755-1

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 2 k 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.