### Abstract

A d-dimensional bar-and-joint framework (G; p) with underlying graph G is called universally rigid if all realizations of G with the same edge lengths, in all dimensions, are congruent to (G; p). We give a complete characterization of universally rigid one-dimensional bar-and-joint frameworks in general position with a complete bipartite underlying graph. We show that the only bipartite graph for which all generic d-dimensional realizations are universally rigid is the complete graph on two vertices, for all d ≥ 1. We also discuss several open questions concerning generically universally rigid graphs and the universal rigidity of general frameworks on the line.

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

Pages (from-to) | 10-21 |

Number of pages | 12 |

Journal | Contributions to Discrete Mathematics |

Volume | 10 |

Issue number | 2 |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

- Bar-and-joint framework
- Bipartite framework
- Cover graph
- Generic rigidity
- Global rigidity
- Universal rigidity

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Contributions to Discrete Mathematics*,

*10*(2), 10-21.

**On universally rigid frameworks on the line.** / Jordán, T.; Nguyen, Viet Hang.

Research output: Contribution to journal › Article

*Contributions to Discrete Mathematics*, vol. 10, no. 2, pp. 10-21.

}

TY - JOUR

T1 - On universally rigid frameworks on the line

AU - Jordán, T.

AU - Nguyen, Viet Hang

PY - 2015

Y1 - 2015

N2 - A d-dimensional bar-and-joint framework (G; p) with underlying graph G is called universally rigid if all realizations of G with the same edge lengths, in all dimensions, are congruent to (G; p). We give a complete characterization of universally rigid one-dimensional bar-and-joint frameworks in general position with a complete bipartite underlying graph. We show that the only bipartite graph for which all generic d-dimensional realizations are universally rigid is the complete graph on two vertices, for all d ≥ 1. We also discuss several open questions concerning generically universally rigid graphs and the universal rigidity of general frameworks on the line.

AB - A d-dimensional bar-and-joint framework (G; p) with underlying graph G is called universally rigid if all realizations of G with the same edge lengths, in all dimensions, are congruent to (G; p). We give a complete characterization of universally rigid one-dimensional bar-and-joint frameworks in general position with a complete bipartite underlying graph. We show that the only bipartite graph for which all generic d-dimensional realizations are universally rigid is the complete graph on two vertices, for all d ≥ 1. We also discuss several open questions concerning generically universally rigid graphs and the universal rigidity of general frameworks on the line.

KW - Bar-and-joint framework

KW - Bipartite framework

KW - Cover graph

KW - Generic rigidity

KW - Global rigidity

KW - Universal rigidity

UR - http://www.scopus.com/inward/record.url?scp=84965000053&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84965000053&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84965000053

VL - 10

SP - 10

EP - 21

JO - Contributions to Discrete Mathematics

JF - Contributions to Discrete Mathematics

SN - 1715-0868

IS - 2

ER -