Connected rigidity matroids and unique realizations of graphs

A d -dimensional framework is a straight line realization of a graph G in R d . 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 R d if every equivalent framework can be obtained from it by an isometry of R d . Bruce Hendrickson proved that if G has a unique realization in R d then G is ( d + 1 ) -connected and redundantly rigid. He conjectured that every realization of a ( d + 1 ) -connected and redundantly rigid graph in R d 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 Hendricksonʹs 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.