### Abstract

A basic geometric question is to determine when a given framework G(p) is globally rigid in Euclidean space Rd, where G is a finite graph and p is a configuration of points corresponding to the vertices of G. G(p) is globally rigid in Rd if for any other configuration q for G such that the edge lengths of G(q) are the same as the corresponding edge lengths of G(p), then p is congruent to q. A framework G(p) is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.When the configuration p is generic, redundant rigidity and ( d+ 1)-connectivity are both necessary conditions for global rigidity. Recent results have shown that for d= 2 and for generic configurations redundant rigidity and 3-connectivity are also sufficient. This gives a good combinatorial characterization in the two-dimensional case that only depends on G and can be checked in polynomial time. It appears that a similar result for d≥. 3 is beyond the scope of present techniques and there are examples showing that the above necessary conditions are not always sufficient.However, there is a special class of generic frameworks that have polynomial time algorithms for their generic rigidity (and redundant rigidity) in Rd for any d≥ 1, namely generic body-and-bar frameworks. Such frameworks are constructed from a finite number of rigid bodies that are connected by bars generically placed with respect to each body. We show that a body-and-bar framework is generically globally rigid in Rd, for any d≥ 1, if and only if it is redundantly rigid. As a consequence there is a deterministic polynomial time combinatorial algorithm to determine the generic global rigidity of body-and-bar frameworks in any dimension.

Original language | English |
---|---|

Pages (from-to) | 689-705 |

Number of pages | 17 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 103 |

Issue number | 6 |

DOIs | |

Publication status | Published - Nov 2013 |

### Fingerprint

### Keywords

- Body-bar framework
- Global rigidity
- Redundant rigidity
- Rigid graph
- Stress matrix

### ASJC Scopus subject areas

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

### Cite this

*Journal of Combinatorial Theory. Series B*,

*103*(6), 689-705. https://doi.org/10.1016/j.jctb.2013.09.002

**Generic global rigidity of body-bar frameworks.** / Connelly, R.; Jordán, T.; Whiteley, W.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 103, no. 6, pp. 689-705. https://doi.org/10.1016/j.jctb.2013.09.002

}

TY - JOUR

T1 - Generic global rigidity of body-bar frameworks

AU - Connelly, R.

AU - Jordán, T.

AU - Whiteley, W.

PY - 2013/11

Y1 - 2013/11

N2 - A basic geometric question is to determine when a given framework G(p) is globally rigid in Euclidean space Rd, where G is a finite graph and p is a configuration of points corresponding to the vertices of G. G(p) is globally rigid in Rd if for any other configuration q for G such that the edge lengths of G(q) are the same as the corresponding edge lengths of G(p), then p is congruent to q. A framework G(p) is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.When the configuration p is generic, redundant rigidity and ( d+ 1)-connectivity are both necessary conditions for global rigidity. Recent results have shown that for d= 2 and for generic configurations redundant rigidity and 3-connectivity are also sufficient. This gives a good combinatorial characterization in the two-dimensional case that only depends on G and can be checked in polynomial time. It appears that a similar result for d≥. 3 is beyond the scope of present techniques and there are examples showing that the above necessary conditions are not always sufficient.However, there is a special class of generic frameworks that have polynomial time algorithms for their generic rigidity (and redundant rigidity) in Rd for any d≥ 1, namely generic body-and-bar frameworks. Such frameworks are constructed from a finite number of rigid bodies that are connected by bars generically placed with respect to each body. We show that a body-and-bar framework is generically globally rigid in Rd, for any d≥ 1, if and only if it is redundantly rigid. As a consequence there is a deterministic polynomial time combinatorial algorithm to determine the generic global rigidity of body-and-bar frameworks in any dimension.

AB - A basic geometric question is to determine when a given framework G(p) is globally rigid in Euclidean space Rd, where G is a finite graph and p is a configuration of points corresponding to the vertices of G. G(p) is globally rigid in Rd if for any other configuration q for G such that the edge lengths of G(q) are the same as the corresponding edge lengths of G(p), then p is congruent to q. A framework G(p) is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.When the configuration p is generic, redundant rigidity and ( d+ 1)-connectivity are both necessary conditions for global rigidity. Recent results have shown that for d= 2 and for generic configurations redundant rigidity and 3-connectivity are also sufficient. This gives a good combinatorial characterization in the two-dimensional case that only depends on G and can be checked in polynomial time. It appears that a similar result for d≥. 3 is beyond the scope of present techniques and there are examples showing that the above necessary conditions are not always sufficient.However, there is a special class of generic frameworks that have polynomial time algorithms for their generic rigidity (and redundant rigidity) in Rd for any d≥ 1, namely generic body-and-bar frameworks. Such frameworks are constructed from a finite number of rigid bodies that are connected by bars generically placed with respect to each body. We show that a body-and-bar framework is generically globally rigid in Rd, for any d≥ 1, if and only if it is redundantly rigid. As a consequence there is a deterministic polynomial time combinatorial algorithm to determine the generic global rigidity of body-and-bar frameworks in any dimension.

KW - Body-bar framework

KW - Global rigidity

KW - Redundant rigidity

KW - Rigid graph

KW - Stress matrix

UR - http://www.scopus.com/inward/record.url?scp=84887616194&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887616194&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2013.09.002

DO - 10.1016/j.jctb.2013.09.002

M3 - Article

VL - 103

SP - 689

EP - 705

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 6

ER -