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 two-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.