The generic rank of body-bar-and-hinge frameworks

Bill Jackson, T. Jordán

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

Tay [T.S. Tay, Rigidity of multi-graphs I Linking Bodies in n-space, J. Combin. Theory B 26 (1984) 95-112] characterized the multigraphs which can be realized as infinitesimally rigid d-dimensional body-and-bar frameworks. Subsequently, Tay [T.S. Tay, Linking (n - 2)-dimensional panels in n-space II: (n - 2, 2)-frameworks and body and hinge structures, Graphs Combin. 5 (1989) 245-273] and Whiteley [W. Whiteley, The union of matroids and the rigidity of frameworks, SIAM J. Discrete Math. 1 (2) (1988) 237-255] independently characterized the multigraphs which can be realized as infinitesimally rigid d-dimensional body-and-hinge frameworks. We adapt Whiteley's proof technique to characterize the multigraphs which can be realized as infinitesimally rigid d-dimensional body-bar-and-hinge frameworks. More importantly, we obtain a sufficient condition for a multigraph to be realized as an infinitesimally rigid d-dimensional body-and-hinge framework in which all hinges lie in the same hyperplane. This result is related to a long-standing conjecture of Tay and Whiteley [T.S. Tay, W. Whiteley, Recent advances in the generic rigidity of structures, Structural Topology 9 (1984) 31-38] which would characterize when a multigraph can be realized as an infinitesimally rigid d-dimensional body-and-hinge framework in which all the hinges incident to each body lie in a common hyperplane. As a corollary we deduce that if a graph G has three spanning trees which use each edge of G at most twice, then its square can be realized as an infinitesimally rigid three-dimensional bar-and-joint framework.

Original languageEnglish
Pages (from-to)574-588
Number of pages15
JournalEuropean Journal of Combinatorics
Volume31
Issue number2
DOIs
Publication statusPublished - Feb 2010

Fingerprint

Hinges
Multigraph
Rigidity
Hyperplane
Linking
Framework
Graph in graph theory
Matroid
Spanning tree
Deduce
Topology
Corollary
Union
Three-dimensional
Sufficient Conditions

ASJC Scopus subject areas

  • Geometry and Topology
  • Theoretical Computer Science
  • Computational Theory and Mathematics

Cite this

The generic rank of body-bar-and-hinge frameworks. / Jackson, Bill; Jordán, T.

In: European Journal of Combinatorics, Vol. 31, No. 2, 02.2010, p. 574-588.

Research output: Contribution to journalArticle

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