Global Rigidity: The Effect of Coning

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, although there is an algorithm [10] 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 for 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 ∗Research supported in part by NSF Grant No. DMS–0209595 (USA). e-mail: [email protected] †Work supported in part by a grant from NSERC (Canada). e-mail: [email protected]