title: Gain-Sparsity and Symmetry-Forced Rigidity in the Plane.
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 2kwith oddk, 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.
authors.affiliation: Department of Operations Research, Eötvös University, and the MTA-ELTE Egerváry Research Group on Combinatorial Optimization, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary.
authors.forename: Tibor
authors.initials: T
authors.lastname: Jordán
authors.affiliation: Department of Operations Research, Eötvös University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary ; Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF UK.
authors.forename: Viktória E
authors.initials: VE
authors.lastname: Kaszanitzky
authors.affiliation: Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto 606-8502 Japan ; Centrum Wiskunde & Informatica (CWI), Postbus 94079, 1090 GB Amsterdam, The Netherlands.
authors.forename: Shin-Ichi
authors.initials: SI
authors.lastname: Tanigawa
keywords: Frame matroids; Frameworks; Group-labeled graphs; Infinitesimal rigidity; Rigidity matroids; Rigidity of graphs; Symmetry