### Abstract

A straight-line realization of (or a bar-and-joint framework on) graph G in ^{Rd} is said to be globally rigid if it is congruent to every other realization of G with the same edge lengths. A graph G is called globally rigid in ^{Rd} if every generic realization of G is globally rigid. We give an algorithm for constructing a globally rigid realization of globally rigid graphs in ^{R2}. If G is triangle-reducible, which is a subfamily of globally rigid graphs that includes Cauchy graphs as well as Grünbaum graphs, the constructed realization will also be infinitesimally rigid. Our algorithm relies on the inductive construction of globally rigid graphs which uses edge additions and one of the Henneberg operations. We also show that vertex splitting, which is another well-known operation in combinatorial rigidity, preserves global rigidity in ^{R2}.

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

Pages (from-to) | 511-521 |

Number of pages | 11 |

Journal | Computational Geometry: Theory and Applications |

Volume | 42 |

Issue number | 6-7 |

DOIs | |

Publication status | Published - Aug 2009 |

### Fingerprint

### Keywords

- Gale transform
- Globally rigid framework
- Rigid graph
- Vertex split

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computer Science Applications
- Computational Mathematics
- Control and Optimization
- Geometry and Topology

### Cite this

*Computational Geometry: Theory and Applications*,

*42*(6-7), 511-521. https://doi.org/10.1016/j.comgeo.2008.09.007

**Operations preserving the global rigidity of graphs and frameworks in the plane.** / Jordán, T.; Szabadka, Zoltán.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 42, no. 6-7, pp. 511-521. https://doi.org/10.1016/j.comgeo.2008.09.007

}

TY - JOUR

T1 - Operations preserving the global rigidity of graphs and frameworks in the plane

AU - Jordán, T.

AU - Szabadka, Zoltán

PY - 2009/8

Y1 - 2009/8

N2 - A straight-line realization of (or a bar-and-joint framework on) graph G in Rd is said to be globally rigid if it is congruent to every other realization of G with the same edge lengths. A graph G is called globally rigid in Rd if every generic realization of G is globally rigid. We give an algorithm for constructing a globally rigid realization of globally rigid graphs in R2. If G is triangle-reducible, which is a subfamily of globally rigid graphs that includes Cauchy graphs as well as Grünbaum graphs, the constructed realization will also be infinitesimally rigid. Our algorithm relies on the inductive construction of globally rigid graphs which uses edge additions and one of the Henneberg operations. We also show that vertex splitting, which is another well-known operation in combinatorial rigidity, preserves global rigidity in R2.

AB - A straight-line realization of (or a bar-and-joint framework on) graph G in Rd is said to be globally rigid if it is congruent to every other realization of G with the same edge lengths. A graph G is called globally rigid in Rd if every generic realization of G is globally rigid. We give an algorithm for constructing a globally rigid realization of globally rigid graphs in R2. If G is triangle-reducible, which is a subfamily of globally rigid graphs that includes Cauchy graphs as well as Grünbaum graphs, the constructed realization will also be infinitesimally rigid. Our algorithm relies on the inductive construction of globally rigid graphs which uses edge additions and one of the Henneberg operations. We also show that vertex splitting, which is another well-known operation in combinatorial rigidity, preserves global rigidity in R2.

KW - Gale transform

KW - Globally rigid framework

KW - Rigid graph

KW - Vertex split

UR - http://www.scopus.com/inward/record.url?scp=84867937834&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867937834&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2008.09.007

DO - 10.1016/j.comgeo.2008.09.007

M3 - Article

VL - 42

SP - 511

EP - 521

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 6-7

ER -