triangulation = list of three-element lists.

Supply a previously computed Delaunay triangulation of P. The input must be a valid Delaunay triangulation in the format returned by ComputationalGeometry[DelaunayTriangulation]: a list of three-element lists of integers, representing triangles in a triangulation of P.

vertices = list of integers, strings or symbols

Specifies the vertices to be used in the generated graph.

weighted = truefalse

If weighted=true, the result is a weighted graph whose edge weights correspond to the Euclidean distance between points. The default is false.

The RelativeNeighborhoodGraph(P,opts) command returns a relative neighbor graph for the point set P.

The parameter P must be a Matrix or list of lists representing a set of points.

Let $P$ be a set of points in $n$ dimensions, let $p$ and $q$ be arbitrary points from $P$, and let $\mathrm{dist}\left(p\,q\right)$ be the Euclidean distance between $p$ and $q$.

The relative neighborhood graph is the undirected graph whose vertices correspond to the points in $P$ and in which vertices $p$ and $q$ share an edge if there does not exist any $r\in P$ such that $\mathrm{dist}\left(p\&comma;r\right)<\mathrm{dist}\left(p\&comma;q\right)$ and $\mathrm{dist}\left(q\&comma;r\right)<\mathrm{dist}\left(p\&comma;q\right)$.

It has the following relationships with other graphs:

The Euclidean minimum spanning tree on P is a subgraph of the relative neighborhood graph on P.

The relative neighborhood graph on P is a subgraph of the Gabriel graph on P.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\&colon;$

$\mathrm{with}\left(\mathrm{GeometricGraphs}\right)\&colon;$

$\mathrm{points}≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(60\&comma;2\&comma;\mathrm{generator}=0..100.\&comma;\mathrm{datatype}=\mathrm{float}\left[8\right]\right)$

$\mathrm{RNG}≔\mathrm{RelativeNeighborhoodGraph}\left(\mathrm{points}\right)$

$\mathrm{RNG}≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 60\; vertices\; and\; 65\; edges}$

$\mathrm{DrawGraph}\left(\mathrm{RNG}\right)$

Supowit, K. J. (1983), "The relative neighborhood graph, with an application to minimum spanning trees", J. ACM, 30 (3): 428–448, doi:10.1145/2402.322386.

The GraphTheory[GeometricGraphs][RelativeNeighborhoodGraph] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.