In mathematics and computational geometry, the Gabriel graph of a set of points in the Euclidean plane expresses one notion of proximity or nearness of those points. Formally, it is the graph with vertex set in which any two distinct points and are adjacent precisely when the closed disc having as a diameter contains no other points. Another way of expressing the same adjacency criterion is that and should be the two closest given points to their midpoint, with no other given point being as close. Gabriel graphs naturally generalize to higher dimensions, with the empty disks replaced by empty closed balls. Gabriel graphs are named after K. Ruben Gabriel, who introduced them in a paper with Robert R. Sokal in 1969.
For Gabriel graphs of infinite random point sets, the finite site percolation threshold gives the fraction of points needed to support connectivity: if a random subset of fewer vertices than the threshold is given, the remaining graph will almost surely have only finite connected components, while if the size of the random subset is more than the threshold, then the remaining graph will almost surely have an infinite component (as well as finite components). This threshold was proved to exist by Bertin, Billiot & Drouilhet (2002), and more precise values of both site and bond thresholds have been given by Norrenbrock.
Related geometric graphs
It is an instance of a beta-skeleton. Like beta-skeletons, and unlike Delaunay triangulations, it is not a geometric spanner: for some point sets, distances within the Gabriel graph can be much larger than the Euclidean distances between points.
- Gabriel, K. Ruben; Sokal, Robert R. (1969), "A new statistical approach to geographic variation analysis", Systematic Biology, 18 (3): 259–278, doi:10.2307/2412323, JSTOR 2412323
- Bertin, Etienne; Billiot, Jean-Michel; Drouilhet, Rémy (2002), "Continuum percolation in the Gabriel graph", Advances in Applied Probability, 34 (4): 689–701, doi:10.1239/aap/1037990948, MR 1938937, S2CID 121288601
- Norrenbrock, Christoph (May 2016), "Percolation threshold on planar Euclidean Gabriel graphs", European Physical Journal B, 89 (5): 111, arXiv:1406.0663, Bibcode:2016EPJB...89..111N, doi:10.1140/epjb/e2016-60728-0, S2CID 254114033
- Matula, David W.; Sokal, Robert R. (1980), "Properties of Gabriel graphs relevant to geographic variation research and clustering of points in the plane", Geographical Analysis, 12 (3): 205–222, doi:10.1111/j.1538-4632.1980.tb00031.x
- Bose, Prosenjit; Devroye, Luc; Evans, William; Kirkpatrick, David (2006), "On the spanning ratio of Gabriel graphs and β-skeletons", SIAM Journal on Discrete Mathematics, 20 (2): 412–427, CiteSeerX 10.1.1.46.4728, doi:10.1137/S0895480197318088, MR 2257270