GraphTheory[GeometricGraphs]
GabrielGraph
construct Gabriel graph
Calling Sequence
Parameters
Options
Description
Definitions
Examples
References
Compatibility
GabrielGraph( P, opts )
P
-
Matrix or list of lists representing set of points
opts
(optional) one or more options as specified below
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 GabrielGraph(P, opts) command returns the Gabriel graph for the point set P.
The parameter P must be a Matrix or list of lists representing a set of points.
Let be a set of points in dimensions, let and be arbitrary points from , and let be the Euclidean distance between and .
The Gabriel graph the graph whose vertices correspond to points in and whose edges consist of those pairs and from for which the closed ball centered halfway between and with diameter equal to contains no other points from .
Formally, define the ball to be those points such that . The Gabriel graph has an edge between and if and only if .
The Gabriel graph has the following relationships with other graphs:
The Euclidean minimum spanning tree on P is a subgraph of the Gabriel graph on P.
The nearest neighbor graph on P is a subgraph of the Gabriel graph on P.
The relative neighborhood graph on P is a subgraph of the Gabriel graph on P.
The Urquhart graph on P is a subgraph of the Gabriel graph on P.
The Gabriel graph on P is a subgraph of the Delaunay graph on P.
Generate a set of random two-dimensional points and draw a Gabriel graph.
Gabriel, Kuno Ruben; Sokal, Robert Reuven (1969), "A new statistical approach to geographic variation analysis", Systematic Biology, Society of Systematic Biologists, 18(3): 259–278. doi:10.2307/2412323.
The GraphTheory[GeometricGraphs][GabrielGraph] command was introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
See Also
DelaunayTriangulation
GeometricGraphs
Download Help Document