Generic Rigidity Matroids with Dilworth Truncations

We prove that the linear matroid that defines 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 ${d+1 \choose 2}$ graphic matroids by applying variants of Dilworth truncation $n_r$ times, where $n_r$ denotes the number of rods. This leads to an alternative proof of Tay's combinatorial characterizations of generic rigidity of rod-bar frameworks and that of identified body-hinge frameworks

Perfect Elimination Orderings for Symmetric Matrices

We introduce a new class of structured symmetric matrices by extending the notion of perfect elimination ordering from graphs to weighted graphs or matrices. This offers a common framework capturing common vertex elimination orderings of monotone families of chordal graphs, Robinsonian matrices and ultrametrics. We give a structural characterization for matrices that admit perfect elimination orderings in terms of forbidden substructures generalizing chordless cycles in graphs.Comment: 16 pages, 3 figure

Linking Rigid Bodies Symmetrically

The mathematical theory of rigidity of body-bar and body-hinge frameworks provides a useful tool for analyzing the rigidity and flexibility of many articulated structures appearing in engineering, robotics and biochemistry. In this paper we develop a symmetric extension of this theory which permits a rigidity analysis of body-bar and body-hinge structures with point group symmetries. The infinitesimal rigidity of body-bar frameworks can naturally be formulated in the language of the exterior (or Grassmann) algebra. Using this algebraic formulation, we derive symmetry-adapted rigidity matrices to analyze the infinitesimal rigidity of body-bar frameworks with Abelian point group symmetries in an arbitrary dimension. In particular, from the patterns of these new matrices, we derive combinatorial characterizations of infinitesimally rigid body-bar frameworks which are generic with respect to a point group of the form $\mathbb{Z}/2\mathbb{Z}\times \dots \times \mathbb{Z}/2\mathbb{Z}$. Our characterizations are given in terms of packings of bases of signed-graphic matroids on quotient graphs. Finally, we also extend our methods and results to body-hinge frameworks with Abelian point group symmetries in an arbitrary dimension. As special cases of these results, we obtain combinatorial characterizations of infinitesimally rigid body-hinge frameworks with $\mathcal{C}_2$ or $\mathcal{D}_2$ symmetry - the most common symmetry groups found in proteins.Comment: arXiv:1308.6380 version 1 was split into two papers. The version 2 of arXiv:1308.6380 consists of Sections 1 - 6 of the version 1. This paper is based on the second part of the version 1 (Sections 7 and 8

A Min-Max . . . Functions and Its Implications

A. Huber and V. Kolmogorov (ISCO 2012) introduced a concept of k-submodular function as a generalization of ordinary submodular (set) functions and bisubmodular functions and obtained a min-max theorem for minimization of k-submodular functions. Also F. Kuivinen (2011) considered submodular functions on (product lattices of) diamonds and showed a min-max theorem for minimization of submodular functions on diamonds. In the present paper we consider a common generalization of k-submodular functions and submodular functions on diamonds, which we call a transversal submodular function (or a t-submodular function, for short). We show a min-max theorem for minimization of t-submodular functions in terms of a new norm composed of ℓ1 and ℓ ∞ norms. This reveals a relationship between the obtained min-max theorem and that for minimization of ordinary submodular set functions due to J. Edmonds (1970). We also show how our min-max theorem for t-submodular functions can be used to prove the min-max theorem for k-submodular functions by Huber and Kolmogorov and that for submodular functions on diamonds by Kuivinen. Moreover, we show a counterexample to a characterization, given by Huber and Kolmogorov (ISCO 2012), of extreme points of the k-submodular polyhedron and make it a correct one by fixing a flaw therein

Infinitesimal rigidity of symmetric bar-joint frameworks

We propose new symmetry-adapted rigidity matrices to analyze the infinitesimal rigidity of bar-joint frameworks of arbitrary-dimension with Abelian point group symmetries. These matrices define new symmetry-adapted rigidity matroids on group-labeled quotient graphs. Using these new tools, we establish combinatorial characterizations of infinitesimally rigid two-dimensional bar-joint frameworks whose joints are positioned as generically as possible subject to the symmetry constraints imposed by a reflection, a half-turn, or a threefold rotation in the plane. For bar-joint frameworks which are generic with respect to any other cyclic point group in the plane, we provide a number of necessary conditions for infinitesimal rigidity