Uniqueness of Low-Rank Matrix Completion by Rigidity Theory
SIAM Journal on Matrix Analysis and Applications
Sensor network localization by eigenvector synchronization over the euclidean group
ACM Transactions on Sensor Networks (TOSN)
Generic global rigidity of body-bar frameworks
Journal of Combinatorial Theory Series B
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Recent results have confirmed that the global rigidity of bar-and-joint frameworks on a graph G is a generic property in Euclidean spaces of all dimensions. Although it is not known if there is a deterministic algorithm that runs in polynomial time and space, to decide if a graph is generically globally rigid, there is an algorithm (Gortler et al. in Characterizing generic global rigidity, arXiv:0710.0907v1, 2007) running in polynomial time and space that will decide with no false positives and only has false negatives with low probability. When there is a framework that is infinitesimally rigid with a stress matrix of maximal rank, we describe it as a certificate which guarantees that the graph is generically globally rigid, although this framework, itself, may not be globally rigid. We present a set of examples which clarify a number of aspects of global rigidity. There is a technique which transfers rigidity to one dimension higher: coning. Here we confirm that the cone on a graph is generically globally rigid in ℝd+1 if and only if the graph is generically globally rigid in ℝd . As a corollary, we see that a graph is generically globally rigid in the d-dimensional sphere $\mathbb{S}^{d}$if and only if it is generically globally rigid in ℝd .