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 points and are adjacent precisely if they are distinct, i.e. , and the closed disc of which line segment is a diameter contains no other elements of . 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.

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 ).

In mathematics and computational geometry, a **Delaunay triangulation** for a given set **P** of discrete points in a plane is a triangulation DT(**P**) such that no point in **P** is inside the circumcircle of any triangle in DT(**P**). Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid sliver triangles. The triangulation is named after Boris Delaunay for his work on this topic from 1934.

In the mathematical field of graph theory, a **spanning tree***T* of an undirected graph *G* is a subgraph that is a tree which includes all of the vertices of *G*, with a minimum possible number of edges. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree. If all of the edges of *G* are also edges of a spanning tree *T* of *G*, then *G* is a tree and is identical to *T*.

The **Euclidean minimum spanning tree** or **EMST** is a minimum spanning tree of a set of *n* points in the plane, where the weight of the edge between each pair of points is the Euclidean distance between those two points. In simpler terms, an EMST connects a set of dots using lines such that the total length of all the lines is minimized and any dot can be reached from any other by following the lines.

A **triangulation of a set of points** in the Euclidean space is a simplicial complex that covers the convex hull of , and whose vertices belong to . In the plane, triangulations are made up of triangles, together with their edges and vertices. Some authors require that all the points of are vertices of its triangulations. In this case, a triangulation of a set of points in the plane can alternatively be defined as a maximal set of non-crossing edges between points of . In the plane, triangulations are special cases of planar straight-line graphs.

A **spatial network** is a graph in which the vertices or edges are *spatial elements* associated with geometric objects, i.e. the nodes are located in a space equipped with a certain metric. The simplest mathematical realization is a lattice or a random geometric graph, where nodes are distributed uniformly at random over a two-dimensional plane; a pair of nodes are connected if the Euclidean distance is smaller than a given neighborhood radius. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks and biological neural networks are all examples where the underlying space is relevant and where the graph's topology alone does not contain all the information. Characterizing and understanding the structure, resilience and the evolution of spatial networks is crucial for many different fields ranging from urbanism to epidemiology.

In computational geometry, the **Theta graph**, or **-graph**, is a type of geometric spanner similar to a Yao graph. The basic method of construction involves partitioning the space around each vertex into a set of *cones*, which themselves partition the remaining vertices of the graph. Like Yao Graphs, a -graph contains at most one edge per cone; where they differ is how that edge is selected. Whereas Yao Graphs will select the nearest vertex according to the metric space of the graph, the -graph defines a fixed ray contained within each cone and selects the nearest neighbor with respect to orthogonal projections to that ray. The resulting graph exhibits several good spanner properties.

**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 2*k* − 3 edges, and such that the whole graph has exactly 2*n* − 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."

**Clique complexes**, **flag complexes**, and **conformal hypergraphs** are closely related mathematical objects in graph theory and geometric topology that each describe the cliques of an undirected graph.

In computational geometry and computer science, the **minimum-weight triangulation** problem is the problem of finding a triangulation of minimal total edge length. That is, an input polygon or the convex hull of an input point set must be subdivided into triangles that meet edge-to-edge and vertex-to-vertex, in such a way as to minimize the sum of the perimeters of the triangles. The problem is NP-hard for point set inputs, but may be approximated to any desired degree of accuracy. For polygon inputs, it may be solved exactly in polynomial time. The minimum weight triangulation has also sometimes been called the **optimal triangulation**.

In computational geometry and geometric graph theory, a ** β-skeleton** or

In graph theory, a ** k-degenerate graph** is an undirected graph in which every subgraph has a vertex of degree at most

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.

A **flip graph** is a graph whose vertices are combinatorial or geometric objects, and whose edges link two of these objects when they can be obtained from one another by an elementary operation called a flip. Flip graphs are special cases of geometric graphs.

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.

- 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 10.1.1.46.4728 , 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: 10.1111/j.1538-4632.1980.tb00031.x . - Norrenbrock, Christoph (2014),
*Percolation threshold on planar Euclidean Gabriel Graphs*, arXiv: 1406.0663 , Bibcode:2014arXiv1406.0663N .

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