# Gabriel graph

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In mathematics and computational geometry, the Gabriel graph of a set ${\displaystyle S}$ of points in the Euclidean plane expresses one notion of proximity or nearness of those points. Formally, it is the graph ${\displaystyle G}$ with vertex set ${\displaystyle S}$ in which any points ${\displaystyle p\in S}$ and ${\displaystyle q\in S}$ are adjacent precisely if they are distinct, i.e. ${\displaystyle p\neq q}$, and the closed disc of which line segment ${\displaystyle {\overline {pq}}}$ is a diameter contains no other elements of ${\displaystyle S}$. 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.

## Percolation

A finite site percolation threshold for Gabriel graphs has been proven to exist by Bertin, Billiot & Drouilhet (2002), and more precise values of both site and bond thresholds have been given by Norrenbrock (2014).

The Gabriel graph is a subgraph of the Delaunay triangulation. It can be found in linear time if the Delaunay triangulation is given ( Matula & Sokal 1980 ).

The Gabriel graph contains, as subgraphs, the Euclidean minimum spanning tree, the relative neighborhood graph, and the nearest neighbor graph.

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 ( Bose et al. 2006 ).

## Related Research Articles

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Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are allowed to be arbitrary continuous curves connecting the vertices, thus it is "the theory of geometric and topological graphs".

In the mathematical field of graph theory, the Erdős–Rényi model is either of two closely related models for generating random graphs or the evolution of a random network. They are named after mathematicians Paul Erdős and Alfréd Rényi, who first introduced one of the models in 1959, while Edgar Gilbert introduced the other model contemporaneously and independently of Erdős and Rényi. In the model of Erdős and Rényi, all graphs on a fixed vertex set with a fixed number of edges are equally likely; in the model introduced by Gilbert, each edge has a fixed probability of being present or absent, independently of the other edges. These models can be used in the probabilistic method to prove the existence of graphs satisfying various properties, or to provide a rigorous definition of what it means for a property to hold for almost all graphs.

A geometric spanner or a t-spanner graph or a t-spanner was initially introduced as a weighted graph over a set of points as its vertices for which there is a t-path between any pair of vertices for a fixed parameter t. A t-path is defined as a path through the graph with weight at most t times the spatial distance between its endpoints. The parameter t is called the stretch factor or dilation factor of the spanner.

The nearest neighbor graph (NNG) for a set of n objects P in a metric space is a directed graph with P being its vertex set and with a directed edge from p to q whenever q is a nearest neighbor of p.

In computational geometry, the relative neighborhood graph (RNG) is an undirected graph defined on a set of points in the Euclidean plane by connecting two points p and q by an edge whenever there does not exist a third point r that is closer to both p and q than they are to each other. This graph was proposed by Godfried Toussaint in 1980 as a way of defining a structure from a set of points that would match human perceptions of the shape of the set.

In graph theory, the Laman graphs are a family of sparse graphs describing the minimally rigid systems of rods and joints in the plane. Formally, a Laman graph is a graph on n vertices such that, for all k, every k-vertex subgraph has at most 2k − 3 edges, and such that the whole graph has exactly 2n − 3 edges. Laman graphs are named after Gerard Laman, of the University of Amsterdam, who in 1970 used them to characterize rigid planar structures. This characterization, however, had already been discovered in 1927 by Hilda Geiringer.

In mesh generation, Ruppert's algorithm, also known as Delaunay refinement, is an algorithm for creating quality Delaunay triangulations. The algorithm takes a planar straight-line graph and returns a conforming Delaunay triangulation of only quality triangles. A triangle is considered poor-quality if it has a circumradius to shortest edge ratio larger than some prescribed threshold. Discovered by Jim Ruppert in the early 1990s, "Ruppert's algorithm for two-dimensional quality mesh generation is perhaps the first theoretically guaranteed meshing algorithm to be truly satisfactory in practice."

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In computational geometry and geometric graph theory, a β-skeleton or beta skeleton is an undirected graph defined from a set of points in the Euclidean plane. Two points p and q are connected by an edge whenever all the angles prq are sharper than a threshold determined from the numerical parameter β.

In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.

A kinetic Euclidean minimum spanning tree is a kinetic data structure that maintains the Euclidean minimum spanning tree (EMST) of a set P of n points that are moving continuously.

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In mathematics and geographic information science, a shortest-path graph is an undirected graph defined from a set of points in the Euclidean plane. The shortest-path graph is proposed with the idea of inferring edges between a point set such that the shortest path taken over the inferred edges will roughly align with the shortest path taken over the imprecise region represented by the point set. The edge set of the shortest-path graph varies based on a single parameter t ≥ 1. When the weight of an edge is defined as its Euclidean length raised to the power of the parameter t ≥ 1, the edge is present in the shortest-path graph if and only if it is the least weight path between its endpoints.

## References

• 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 .
• 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  , doi:10.1137/S0895480197318088, MR   2257270 .
• 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, JSTOR   2412323 .
• Matula, David W.; Sokal, Robert Reuven (1980), "Properties of Gabriel graphs relevant to geographic variation research and clustering of points in the plane", Geogr. Anal., 12 (3): 205–222, doi:.
• Norrenbrock, Christoph (2014), Percolation threshold on planar Euclidean Gabriel Graphs, arXiv:, Bibcode:2014arXiv1406.0663N .