A proof of Connelly's conjecture on 3-connected circuits of the rigidity matroid

Alex R. Berg, Tibor Jordán

Research output: Contribution to journalArticle

60 Citations (Scopus)

Abstract

A graph G = (V, E) is called a generic circuit if E = 2 V - 2 and every X ⊂ V with 2≤ X ≤ V - 1 satisfies i(X)≤2 X - 3. Here i(X) denotes the number of edges induced by X. The operation extension subdivides an edge uw of a graph by a new vertex v and adds a new edge vz for some vertex z ≠ u, w. Connelly conjectured that every 3-connected generic circuit can be obtained from K4 by a sequence of extensions. We prove this conjecture. As a corollary, we also obtain a special case of a conjecture of Hendrickson on generically globally rigid graphs.

Original languageEnglish
Pages (from-to)77-97
Number of pages21
JournalJournal of Combinatorial Theory. Series B
Volume88
Issue number1
DOIs
Publication statusPublished - May 1 2003

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ASJC Scopus subject areas

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

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