### Abstract

A two-dimensional direction-length framework is a pair (G, p), where G = (V; D, L) is a graph whose edges are labeled as 'direction' or 'length' edges, and a map p from V to ^{2}. The label of an edge uv represents a direction or length constraint between p(u) and p(v). The framework (G, p) is called globally rigid if every other framework (G, q) in which the direction or length between the endvertices of corresponding edges is the same, is 'congruent' to (G, p), i.e. it can be obtained from (G, p) by a translation and, possibly, a dilation by -1. We show that labeled versions of the two Henneberg operations (0-extension and 1-extension) preserve global rigidity of generic direction-length frameworks. These results, together with appropriate inductive constructions, can be used to verify global rigidity of special families of generic direction-length frameworks.

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

Pages (from-to) | 685-706 |

Number of pages | 22 |

Journal | International Journal of Computational Geometry and Applications |

Volume | 20 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 2010 |

### Fingerprint

### Keywords

- direction and length constraints
- Globally rigid frameworks
- rigid graphs

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Applied Mathematics
- Geometry and Topology
- Computational Mathematics

### Cite this

**Operations preserving global rigidity of generic direction-length frameworks.** / Jackson, Bill; Jordán, T.

Research output: Contribution to journal › Article

*International Journal of Computational Geometry and Applications*, vol. 20, no. 6, pp. 685-706. https://doi.org/10.1142/S0218195910003487

}

TY - JOUR

T1 - Operations preserving global rigidity of generic direction-length frameworks

AU - Jackson, Bill

AU - Jordán, T.

PY - 2010/12

Y1 - 2010/12

N2 - A two-dimensional direction-length framework is a pair (G, p), where G = (V; D, L) is a graph whose edges are labeled as 'direction' or 'length' edges, and a map p from V to 2. The label of an edge uv represents a direction or length constraint between p(u) and p(v). The framework (G, p) is called globally rigid if every other framework (G, q) in which the direction or length between the endvertices of corresponding edges is the same, is 'congruent' to (G, p), i.e. it can be obtained from (G, p) by a translation and, possibly, a dilation by -1. We show that labeled versions of the two Henneberg operations (0-extension and 1-extension) preserve global rigidity of generic direction-length frameworks. These results, together with appropriate inductive constructions, can be used to verify global rigidity of special families of generic direction-length frameworks.

AB - A two-dimensional direction-length framework is a pair (G, p), where G = (V; D, L) is a graph whose edges are labeled as 'direction' or 'length' edges, and a map p from V to 2. The label of an edge uv represents a direction or length constraint between p(u) and p(v). The framework (G, p) is called globally rigid if every other framework (G, q) in which the direction or length between the endvertices of corresponding edges is the same, is 'congruent' to (G, p), i.e. it can be obtained from (G, p) by a translation and, possibly, a dilation by -1. We show that labeled versions of the two Henneberg operations (0-extension and 1-extension) preserve global rigidity of generic direction-length frameworks. These results, together with appropriate inductive constructions, can be used to verify global rigidity of special families of generic direction-length frameworks.

KW - direction and length constraints

KW - Globally rigid frameworks

KW - rigid graphs

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U2 - 10.1142/S0218195910003487

DO - 10.1142/S0218195910003487

M3 - Article

VL - 20

SP - 685

EP - 706

JO - International Journal of Computational Geometry and Applications

JF - International Journal of Computational Geometry and Applications

SN - 0218-1959

IS - 6

ER -