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.
Highlights
A d-dimensional bar-and-joint framework is a straight-line realization of a finite simple graph G in Euclidean d-space
We think of a bar-and-joint framework as a collection of fixed-length bars which are connected at their ends by universal joints
The works initiated by Ross [17] and Malestein and Theran [11] gave natural extensions of Laman’s theorem to periodic frameworks in the plane, where the ingenious idea is to look at count conditions for quotient graphs with group labelings
Summary
A d-dimensional bar-and-joint framework (or, a framework) is a straight-line realization of a finite simple graph G in Euclidean d-space. By using the orbit rigidity matrix introduced by Schulze and Whiteley [23], we can reformulate our problem in terms of the generic rank of a matrix in which each row corresponds to an edge orbit and each vertex orbit has two columns This in turn is equivalent to characterizing independence in a matroid defined on the edge set of the group-labeled quotient graph, in which vertices and edges correspond to vertex and edge orbits, respectively, and which concisely represents the graph structure with the. Most of our inductive steps (extending or reducing a symmetric framework or a labeled graph, respectively) are valid for dihedral groups D2k with even k, and can be used to show that in the even case the irreducible graphs (frameworks), where our reduction operations are not applicable, are very special The smallest such framework, which is predicted to be rigid by the matroidal count but is flexible is the Bottema mechanism, a well-known mechanism in the kinematics literature (see, e.g., [30]). For a cyclic subgroup C of D, Cdenotes the maximal cyclic subgroup containing C
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