### 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 language | English |
---|---|

Journal | Journal of Combinatorial Theory. Series B |

DOIs | |

Publication status | Accepted/In press - Jul 19 2014 |

### Fingerprint

### 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

*Journal of Combinatorial Theory. Series B*. https://doi.org/10.1016/j.jctb.2015.11.003

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

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*. https://doi.org/10.1016/j.jctb.2015.11.003

}

TY - JOUR

T1 - Generic global rigidity of body-hinge frameworks

AU - Jordán, T.

AU - Király, Csaba

AU - Tanigawa, Shin ichi

PY - 2014/7/19

Y1 - 2014/7/19

N2 - 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.

AB - 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.

KW - Body-hinge framework

KW - Edge-disjoint spanning trees

KW - Global rigidity

KW - Redundant rigidity

KW - Rigid graph

UR - http://www.scopus.com/inward/record.url?scp=84949058040&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84949058040&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2015.11.003

DO - 10.1016/j.jctb.2015.11.003

M3 - Article

AN - SCOPUS:84949058040

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

ER -