Gabriel graph

Gabriel graph

A Gabriel graph is a certain graph that connects a set of points in the Euclidean plane. Two points "P" and "Q" are connected by an edge in the Gabriel graph whenever the disk having line segment "PQ" as its diameter contains no other points from the given point set. More generally, in any dimension, the Gabriel graph connects any two points forming the endpoints of the diameter of an empty sphere. Gabriel graphs are named after K. R. Gabriel, who introduced them in a paper with R. R. Sokal in 1969.

The Gabriel graph is a subgraph of the Delaunay triangulation; it can be found in linear time if the Delaunay triangulation is given (Matula and Sokal, 1980). The Gabriel graph contains as a subgraph the Euclidean minimum spanning tree and the nearest neighbor graph. It is an instance of a beta-skeleton.

References

*citation
last1 = Gabriel | first1 = K. R. | last2 = Sokal | first2 = R. R.
title = A new statistical approach to geographic variation analysis
journal = Systematic Zoology
year = 1969
pages = 259–270
volume = 18
doi = 10.2307/2412323
.

*citation
last1 = Matula | first1 = D. W. | last2 = Sokal | first2 = R. R.
title = Properties of Gabriel graphs relevant to geographic variation research and clustering of points in the plane
journal = Geogr. Anal.
year = 1980
pages = 205–222
volume = 12
.


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