The Union of Matroids and the Rigidity of Frameworks

Abstract

From the pattern of its rigidity matrix, we show that a k-frame on a graph (or multigraph) has the matroid structure of the union of k copies of the cycle matroid of the graph. This matrix pattern is applied to three central results about the rigidity of frameworks. An immediate corollary of this matroid union is a characterization of rigid bar and body frameworks in n-space (Tay’s Theorem). This is further specialized to characterize the independence and the rigidity of body and hinge structures in n-space (a new theorem). The two-frame, or union of two copies of the graphic matroid, is truncated to produce plane bar and joint frameworks giving a characterization of minimal infinitesimally rigid bar and joint frameworks in the plane (Laman’s Theorem). Finally, these techniques are used to characterize the graphs of infinitesimally rigid frameworks on other surfaces, such as the flat torus, the cylinder, cones, etc., using matroid unions of cycle and bicycle matroids of the graph.