Relative neighborhood graph

In computational geometry, the relative neighborhood graph (RNG) is an undirected graph defined on a set of points in the Euclidean plane by connecting two points ${\displaystyle p}$ and ${\displaystyle q}$ by an edge whenever there does not exist a third point ${\displaystyle r}$ that is closer to both ${\displaystyle p}$ and ${\displaystyle q}$ than they are to each other. This graph was proposed by Godfried Toussaint in 1980 as a way of defining a structure from a set of points that would match human perceptions of the shape of the set.^[1][2]

Supowit (1983) showed how to construct the relative neighborhood graph of ${\displaystyle n}$ points in the plane efficiently in ${\displaystyle O(n\log n)}$ time.^[3] It can be computed in ${\displaystyle O(n)}$ expected time, for random set of points distributed uniformly in the unit square.^[4] The relative neighborhood graph can be computed in linear time from the Delaunay triangulation of the point set.^[5][6]

Because it is defined only in terms of the distances between points, the relative neighborhood graph can be defined for point sets in any dimension,^[1][7][8] and for non-Euclidean metrics.^{[1][5][9][10]} Computing the relative neighborhood graph, for higher-dimensional point sets, can be done in time ${\displaystyle O(n^{2})}$.

The relative neighborhood graph is an example of a lens-based beta skeleton. It is a subgraph of the Delaunay triangulation. In turn, the Euclidean minimum spanning tree is a subgraph of it, from which it follows that it is a connected graph.

The Urquhart graph, the graph formed by removing the longest edge from every triangle in the Delaunay triangulation, was originally proposed as a fast method to compute the relative neighborhood graph.^[11] Although the Urquhart graph sometimes differs from the relative neighborhood graph^[12] it can be used as an approximation to the relative neighborhood graph.^[13]