Generic global rigidity of body-bar frameworks

A basic geometric question is to determine when a given framework G(p) is globally rigid in Euclidean space Rd, where G is a finite graph and p is a configuration of points corresponding to the vertices of G. G(p) is globally rigid in Rd if for any other configuration q for G such that the edge lengths of G(q) are the same as the corresponding edge lengths of G(p), then p is congruent to q. A framework G(p) is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G. Hendrickson [9] proved that redundant rigidity is a necessary condition for generic global rigidity, as is (d+1)-connectivity. Recent results in [2] and [10] have shown that when the configuration p is generic and d = 2, redundant rigidity and 3-connectivity are also sufficient a good combinatorial characterization that only depends on G and can be checked in polynomial time. It appears that a similar result for d = 3 is beyond the scope of present techniques ∗Department of Mathematics, Cornell University, Ithaca, NY 14853, USA. Research supported in part by NSF Grant No. DMS–0209595 (USA). e-mail: [email protected] †Department of Operations Research, Eötvös University, Pázmány sétány 1/C, 1117 Budapest, Hungary. Supported by the MTA-ELTE Egerváry Research Group on Combinatorial Optimization and the Hungarian Scientific Research Fund grant no. T49671, T60802. e-mail: [email protected] ‡Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada. Research supported in part by a grant from NSERC (Canada). e-mail: [email protected]