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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics