Generic Rigidity Matroids with Dilworth Truncations

Shin-ichi Tanigawa
2012 SIAM Journal on Discrete Mathematics  
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 2 copies of a graphic matroid by applying variants of Dilworth truncation operations nr times, where nr 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
more » ... identified body-hinge frameworks. Two-dimensional bar-joint framework, (b) Body-bar framework, (c) Body-hinge framework, and (d) Rod-bar framework. trees. Jackson and Jordán [14] further discuss the relation of generic rigidity of the body-bar-hinge model to the forest-packing problem in undirected graphs. Although it is barely mentioned, Tay's work was actually done in a more general setting. An identified body-hinge framework is a body-hinge framework in which each hinge is allowed to connect more than two bodies. Historically, a combinatorial characterization of identified body-hinge frameworks was first conjectured by Tay and Whiteley in [34] , and Tay affirmatively solved the conjecture in [31] as a by-product of his combinatorial characterization of rod-bar frameworks. A rod-bar framework is a structure consisting of disjoint rods linked by bars in R 3 (Figure 1.1(d) ). Each bar connects two rods, and each rod is allowed to be incident to several distinct bars. This structural model naturally comes up from body-bar frameworks by regarding each rod as a degenerated one-dimensional body. Unfortunately, Tay's proof is based on a Henneberg-type graph construction with intricate and long analysis (the combinatorial part now follows from the recent result by Frank and Szegö [6]), and the combinatorics behind rigidity of rod-bar frameworks has not been understood well. To shed light on Tay's result, this paper provides a new proof of the combinatorial characterization of rod-bar frameworks. We cope with a more general structural model, body-rod-bar frameworks, and prove that the linear matroid defining its generic rigidity is equal to a counting matroid defined on the underlying graphs (Theorem 4.9 and Corollary 4.14). Our proof technique is inspired by the idea of Lovász and Yemini given in [21] . They proved, as a new proof of Laman's theorem, that the linear matroid that defines the generic rigidity of two-dimensional bar-joint frameworks can be obtained from the union of two copies of a graphic matroid by Dilworth truncation. Roughly speaking, Dilworth truncation is an operation to construct a new linear matroid from an old one by restricting the domain of entries of each vector to a generic hyperplane. (See subsection 2.4 for the definition.) The main difference between our situation and that of Lovász and Yemini is that we need to apply such truncation operations more than once (while they used it only once). Indeed, it is not trivial to keep up the representation of the resulting matroid when applying Dilworth truncation operations several times, as each hyperplane must be inserted in a "generic" position relative to the preceding hyperplanes. We will overcome the difficulty by extending an idea of Lovász [20] so that each truncation is performed within a designated subspace. A bar-joint framework can be considered as a body-bar framework consisting of zero-dimensional bodies. As combinatorial properties of body-bar frameworks with three-dimensional bodies are well understood [30, 37, 39] in R 3 , it is then natural to consider body-bar frameworks with one-dimensional bodies (i.e., rods) toward a combinatorial characterization of bar-joint frameworks. Our proof explicitly describes how each three-dimensional body can be replaced by a one-dimensional body by the use of truncations.
doi:10.1137/100819473 fatcat:55w5jkgi3baf7n3avnrepx5me4