GabrielGraph - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


GraphTheory[GeometricGraphs]

  

GabrielGraph

  

construct Gabriel graph

 

Calling Sequence

Parameters

Options

Description

Definitions

Examples

References

Compatibility

Calling Sequence

GabrielGraph( P, opts )

Parameters

P

-

Matrix or list of lists representing set of points

opts

-

(optional) one or more options as specified below

Options

• 

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.

Description

• 

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.

Definitions

• 

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.

Examples

Generate a set of random two-dimensional points and draw a Gabriel graph.

(1)

References

  

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.

Compatibility

• 

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