The relative neighbourhood graph of a finite planar set

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. Relative neighbourhood graph Minimal spanning tree Triangulations Delaunay triangulation Dot patterns Computational perception Pattern recognition Algorithms Geometric complexity Geometric probability CR Categories: 3.36 3.63 5.25 5.32 5.5