Global Rigidity: An algorithm and 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. While this creates the possibility of a combinatorial characterization of genericaly globally rigid graphs, the only current combinatorial characterizations, which are known to be deterministically computable in polynomial time, are in dimensions 1 and 2. We present a certificate for a framework which guarantees that the graph is generically globally rigid, and a corresponding algorithm, with polynomial complexity bound in the size of the graph, that can determine global rigidity for a given graph G. The algorithm has no false positives. In other words, if it reports that G is globally rigid, it is, but it may incorrectly report that G is probably not globally rigid. With this algorithm, we present a set of examples which clarify ∗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]