Connected rigidity matroids and unique realizations of graphs

Bill Jackson, Tibor Jordán

Research output: Contribution to journalArticle

219 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1-29
Number of pages29
JournalJournal of Combinatorial Theory. Series B
Volume94
Issue number1
DOIs
Publication statusPublished - May 1 2005

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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