### Abstract

The relative neighbourhood graph (RNG) of a set of n points on the plane is defined. The ability of the RNG to extract a perceptually meaningful structure from the set of points is briefly discussed and compared to that of two other graph structures: the minimal spanning tree (MST) and the Delaunay (Voronoi) triangulation (DT). It is shown that the RNG is a superset of the MST and a subset of the DT. Two algorithms for obtaining the RNG of n points on the plane are presented. One algorithm runs in 0(n^{2}) time and the other runs in 0(n^{3}) time but works also for the d-dimensional case. Finally, several open problems concerning the RNG in several areas such as geometric complexity, computational perception, and geometric probability, are outlined.

Original language | English (US) |
---|---|

Pages (from-to) | 261-268 |

Number of pages | 8 |

Journal | Pattern Recognition |

Volume | 12 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1980 |

### Fingerprint

### Keywords

- Algorithms
- Computational perception
- Delaunay triangulation
- Dot patterns
- Geometric complexity Geometric probability
- Minimal spanning tree
- Pattern recognition
- Relative neighbourhood graph
- Triangulations

### ASJC Scopus subject areas

- Software
- Signal Processing
- Computer Vision and Pattern Recognition
- Artificial Intelligence

### Cite this

**The relative neighbourhood graph of a finite planar set.** / Toussaint, Godfried.

Research output: Contribution to journal › Article

*Pattern Recognition*, vol. 12, no. 4, pp. 261-268. https://doi.org/10.1016/0031-3203(80)90066-7

}

TY - JOUR

T1 - The relative neighbourhood graph of a finite planar set

AU - Toussaint, Godfried

PY - 1980/1/1

Y1 - 1980/1/1

N2 - The relative neighbourhood graph (RNG) of a set of n points on the plane is defined. The ability of the RNG to extract a perceptually meaningful structure from the set of points is briefly discussed and compared to that of two other graph structures: the minimal spanning tree (MST) and the Delaunay (Voronoi) triangulation (DT). It is shown that the RNG is a superset of the MST and a subset of the DT. Two algorithms for obtaining the RNG of n points on the plane are presented. One algorithm runs in 0(n2) time and the other runs in 0(n3) time but works also for the d-dimensional case. Finally, several open problems concerning the RNG in several areas such as geometric complexity, computational perception, and geometric probability, are outlined.

AB - The relative neighbourhood graph (RNG) of a set of n points on the plane is defined. The ability of the RNG to extract a perceptually meaningful structure from the set of points is briefly discussed and compared to that of two other graph structures: the minimal spanning tree (MST) and the Delaunay (Voronoi) triangulation (DT). It is shown that the RNG is a superset of the MST and a subset of the DT. Two algorithms for obtaining the RNG of n points on the plane are presented. One algorithm runs in 0(n2) time and the other runs in 0(n3) time but works also for the d-dimensional case. Finally, several open problems concerning the RNG in several areas such as geometric complexity, computational perception, and geometric probability, are outlined.

KW - Algorithms

KW - Computational perception

KW - Delaunay triangulation

KW - Dot patterns

KW - Geometric complexity Geometric probability

KW - Minimal spanning tree

KW - Pattern recognition

KW - Relative neighbourhood graph

KW - Triangulations

UR - http://www.scopus.com/inward/record.url?scp=0019213414&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0019213414&partnerID=8YFLogxK

U2 - 10.1016/0031-3203(80)90066-7

DO - 10.1016/0031-3203(80)90066-7

M3 - Article

VL - 12

SP - 261

EP - 268

JO - Pattern Recognition

JF - Pattern Recognition

SN - 0031-3203

IS - 4

ER -