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GraphTheory[GeometricGraphs]

  

UrquhartGraph

  

construct Urquhart graph

 

Calling Sequence

Parameters

Options

Description

Definitions

Examples

References

Compatibility

Calling Sequence

UrquhartGraph( 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 : true or false

  

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 UrquhartGraph(P, opts) command returns an Urquhart graph for the point set P.

• 

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

Definitions

• 

Let P be a set of points in n dimensions.

• 

The Urquhart graph is the undirected graph whose vertices correspond to points in P and in which an edge between p and q exists if p and q are the edge of a triangle in a Delaunay triangulation of P but are not the longest of any triangle in this triangulation.

• 

Intuitively, the Urquhart graph is obtained from a Delaunay triangulation by simply removing the longest edge from each triangle. It was proposed by R. B. Urquhart as an efficient method for approximating the relative neighborhood graph.

• 

The Urquhart graph has the following relationships with other graphs:

  

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

  

The Urquhart graph on P is a subgraph of the Gabriel graph on P.

Examples

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

withGraphTheory:

withGeometricGraphs:

pointsLinearAlgebra:-RandomMatrix60,2,generator=0..100.,datatype=float8

(1)

UGUrquhartGraphpoints

UGGraph 1: an undirected unweighted graph with 60 vertices and 65 edge(s)

(2)

DrawGraphUG

References

  

Urquhart, R. B. (1980), "Algorithms for computation of relative neighborhood graph", Electronics Letters, 16 (14): 556–557. doi:10.1049/el:19800386

Compatibility

• 

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

• 

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

See Also

DelaunayTriangulation

GeometricGraphs