Generic Rigidity Matroids with Dilworth Truncations

Abstract

We prove that the linear matroid that defines the generic rigidity of $d$-dimensional body-rod-bar frameworks (i.e., structures consisting of disjoint bodies and rods mutually linked by bars) can be obtained from the union of ${d+1 \choose 2}$ copies of a graphic matroid by applying variants of Dilworth truncation operations $n_r$ times, where $n_r$ denotes the number of rods. This result leads to an alternative proof of Tay's combinatorial characterizations of the generic rigidity of rod-bar frameworks and that of identified body-hinge frameworks.