8,072 research outputs found
On the universal rigidity of generic bar frameworks
Let be a finite set. An -configuration is a mapping , where
are not contained in a proper hyper-plane. A framework in is an -configuration together with a graph such that every two points corresponding to adjacent vertices of are constrained to stay the same distance apart. A framework is said to be generic if all the coordinates of are algebraically independent over the integers. A framework in is said to be unique if there does not exist a framework in , for some , , such that for all . In this paper we present a sufficient condition for a generic framework to be unique, and we conjecture that this condition is also necessary. Connections with the closely related problems of global rigidity and dimensional rigidity are also discussed
On universally rigid frameworks on the line
A -dimensional bar-and-joint framework with underlying graph is called universally rigid if all realizations of with the same edge lengths, in all dimensions, are congruent to . We give a complete characterization of universally rigid one-dimensional bar-and-joint frameworks in general position with a complete bipartite underlying graph. We show that the only bipartite graph for which all generic -dimensional realizations are universally rigid is the complete graph on two vertices, for all . We also discuss several open questions concerning generically universally rigid graphs and the universal rigidity of general frameworks on the line.
Iterative Universal Rigidity
A bar framework determined by a finite graph and configuration in
space is universally rigid if it is rigid in any . We provide a characterization of universally rigidity for any
graph and any configuration in terms of a sequence of affine
subsets of the space of configurations. This corresponds to a facial reduction
process for closed finite dimensional convex cones.Comment: 41 pages, 12 figure
Characterizing the universal rigidity of generic frameworks
A framework is a graph and a map from its vertices to E^d (for some d). A
framework is universally rigid if any framework in any dimension with the same
graph and edge lengths is a Euclidean image of it. We show that a generic
universally rigid framework has a positive semi-definite stress matrix of
maximal rank. Connelly showed that the existence of such a positive
semi-definite stress matrix is sufficient for universal rigidity, so this
provides a characterization of universal rigidity for generic frameworks. We
also extend our argument to give a new result on the genericity of strict
complementarity in semidefinite programming.Comment: 18 pages, v2: updates throughout; v3: published versio
Graph connectivity and universal rigidity of bar frameworks
Let be a graph on nodes. In this note, we prove that if is
-vertex connected, , then there exists a
configuration in general position in such that the bar framework
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
Natural realizations of sparsity matroids
A hypergraph G with n vertices and m hyperedges with d endpoints each is
(k,l)-sparse if for all sub-hypergraphs G' on n' vertices and m' edges, m'\le
kn'-l. For integers k and l satisfying 0\le l\le dk-1, this is known to be a
linearly representable matroidal family.
Motivated by problems in rigidity theory, we give a new linear representation
theorem for the (k,l)-sparse hypergraphs that is natural; i.e., the
representing matrix captures the vertex-edge incidence structure of the
underlying hypergraph G.Comment: Corrected some typos from the previous version; to appear in Ars
Mathematica Contemporane
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
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