Generic global rigidity of body-hinge frameworks

T. Jordán, Csaba Király, Shin ichi Tanigawa

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

A d-dimensional body-hinge framework is a structure consisting of rigid bodies in d-space in which some pairs of bodies are connected by a hinge, restricting the relative position of the corresponding bodies. The framework is said to be globally rigid if every other arrangement of the bodies and their hinges can be obtained by a congruence of the space. The combinatorial structure of a body-hinge framework can be encoded by a multigraph H, in which the vertices correspond to the bodies and the edges correspond to the hinges. We prove that a generic body-hinge realization of a multigraph H is globally rigid in Rd, d≥3, if and only if ((d+12)-1)H-e contains (d+12) edge-disjoint spanning trees for all edges e of ((d+12)-1)H. (For a multigraph H and integer k we use kH to denote the multigraph obtained from H by replacing each edge e of H by k parallel copies of e.) This implies an affirmative answer to a conjecture of Connelly, Whiteley, and the first author.We also consider bar-joint frameworks and show, for each d≥3, an infinite family of graphs satisfying Hendrickson's well-known necessary conditions for generic global rigidity in Rd (that is, (d+1)-connectivity and redundant rigidity) which are not generically globally rigid in Rd. The existence of these families disproves a number of conjectures, due to Connelly, Connelly and Whiteley, and the third author, respectively.

Original languageEnglish
JournalJournal of Combinatorial Theory. Series B
DOIs
Publication statusAccepted/In press - Jul 19 2014

Fingerprint

Multigraph
Hinges
Rigidity
Disprove
D-space
Spanning tree
Rigid Body
Congruence
Arrangement
Disjoint
Connectivity
If and only if
Denote
Imply
Necessary Conditions
Integer
Framework
Graph in graph theory
Family

Keywords

  • Body-hinge framework
  • Edge-disjoint spanning trees
  • Global rigidity
  • Redundant rigidity
  • Rigid graph

ASJC Scopus subject areas

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

Cite this

Generic global rigidity of body-hinge frameworks. / Jordán, T.; Király, Csaba; Tanigawa, Shin ichi.

In: Journal of Combinatorial Theory. Series B, 19.07.2014.

Research output: Contribution to journalArticle

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