Graph connectivity and universal rigidity of bar frameworks

Let $G$ be a graph on $n$ nodes. In this note, we prove that if $G$ is $(r+1)$-vertex connected, $1 \leq r \leq n-2$, then there exists a configuration $p$ in general position in $R^r$ such that the bar framework $(G,p)$ is universally rigid. The proof is constructive and is based on a theorem by Lovasz et al concerning orthogonal representations and connectivity of graphs [12,13].Comment: updated versio

On Farkas Lemma and Dimensional Rigidity of Bar Frameworks

We present a new semidefinite Farkas lemma involving a side constraint on the rank. This lemma is then used to present a new proof of a recent characterization, by Connelly and Gortler, of dimensional rigidity of bar frameworks.Comment: First Draf

Toward the Universal Rigidity of General Frameworks

Let (G,P) be a bar framework of n vertices in general position in R^d, d <= n-1, where G is a (d+1)-lateration graph. In this paper, we present a constructive proof that (G,P) admits a positive semi-definite stress matrix with rank n-d-1. We also prove a similar result for a sensor network where the graph consists of m(>= d+1) anchors.Comment: v2, a revised version of an earlier submission (v1

On dimensional rigidity of bar-and-joint frameworks

AbstractLet V={1,2,…,n}. A mapping p:V→Rr, where p1,…,pn are not contained in a proper hyper-plane is called an r-configuration. Let G=(V,E) be a simple connected graph on n vertices. Then an r-configuration p together with graph G, where adjacent vertices of G are constrained to stay the same distance apart, is called a bar-and-joint framework (or a framework) in Rr, and is denoted by G(p). In this paper we introduce the notion of dimensional rigidity of frameworks, and we study the problem of determining whether or not a given G(p) is dimensionally rigid. A given framework G(p) in Rr is said to be dimensionally rigid iff there does not exist a framework G(q) in Rs for s⩾r+1, such that ∥qi-qj∥2=∥pi-pj∥2 for all (i,j)∈E. We present necessary and sufficient conditions for G(p) to be dimensionally rigid, and we formulate the problem of checking the validity of these conditions as a semidefinite programming (SDP) problem. The case where the points p1,…,pn of the given r-configuration are in general position, is also investigated