Connected rigidity matroids and unique realizations of graphs

A d-dimensional framework is a straight line realization of a graph G in Rd . We shall only consider generic frameworks, in which the co-ordinates of all the vertices of G are algebraically independent. Two frameworks for G are equivalent if corresponding edges in the two frameworks have the same length. A framework is a unique realization of G in Rd if every equivalent framework can be obtained from it by an isometry of Rd . Bruce Hendrickson proved that if G has a unique realization in Rd then G is (d + 1)-connected and redundantly rigid. He conjectured that every realization of a (d + 1)connected and redundantly rigid graph in Rd is unique. This conjecture is true for d = 1 but was disproved by Robert Connelly for d ≥ 3. We resolve the remaining open case by showing that Hendrickson’s conjecture is true for d = 2. As a corollary we deduce that every realization of a 6-connected graph as a 2-dimensional generic framework is a unique realization. Our proof is based on a new inductive characterization of 3-connected graphs whose rigidity matroid is connected. ∗Supported by the Royal Society/Hungarian Academy of Sciences Exchange Programme. †Supported by the MTA-ELTE Egerváry Research Group on Combinatorial Optimization, and the Hungarian Scientific Research Fund grant No. F034930, T037547, and FKFP grant No. 0143/2001.