On stress matrices of chordal bar frameworks in general position

Abstract

A bar framework in R^r, denoted by G(p), is a simple connected graph G whose vertices are points p^1,...,p^n in R^r that affinely span R^r, and whose edges are line segments between pairs of these points. In this paper, we use stress matrices to characterize the universal and global rigidities of chordal bar frameworks in general position in R^r, i.e., bar frameworks where graph G is chordal and the points p^1,...,p^n are in general position in R^r. We also prove that if a chordal bar framework in R^r admits a stress matrix of rank n-r-1 with generic rank profile, then it admits a positive semidefinite stress matrix of rank n-r-1.