Gabriel graph

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The Gabriel graph of 100 random points Gabriel graph.svg
The Gabriel graph of 100 random points
Gabriel Pairs.svg
Points and are Gabriel neighbours, as is outside their diameter circle.
Not Gabriel Pairs.svg
The presence of point within the circle prevents points and from being Gabriel neighbors.

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. [1]

Contents

Percolation

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), [2] and more precise values of both site and bond thresholds have been given by Norrenbrock. [3]

The Gabriel graph is a subgraph of the Delaunay triangulation. It can be found in linear time if the Delaunay triangulation is given. [4]

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. [5]

Related Research Articles

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In mathematics and computational geometry, a Delaunay triangulation for a given set P of discrete points in a general position 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.

Percolation theory Mathematical theory on behavior of connected clusters in a random graph

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Component (graph theory) Maximal subgraph whose vertices can reach each other

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This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.

Random graph Graph generated by a random process

In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a random graph.

Spanning tree Tree which includes all vertices of a graph

In the mathematical field of graph theory, a spanning treeT of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. 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.

Euclidean minimum spanning tree

The Euclidean minimum spanning tree or EMST is a minimum spanning tree of a set of points in the plane or higher-dimensional Euclidean space. It connects the points by a system of line segments, so that any two points can reach each other along a path through the line segments, and it selects line segments that minimize the sum of the Euclidean distances between directly-connected pairs of points.

Paley graph

In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally.

Rado graph Infinite graph containing all countable graphs

In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of Wilhelm Ackermann (1937). The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.

Erdős–Rényi model Two closely related models for generating random 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 Hungarian 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, also called the Erdős–Rényi–Gilbert model, 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.

Nearest neighbor graph Type of directed graph

The nearest neighbor graph (NNG) is a directed graph defined for a set of points in a metric space, such as the Euclidean distance in the plane. The NNG has a vertex for each point, and a directed edge from p to q whenever q is a nearest neighbor of p, a point whose distance from p is minimum among all the given points other than p itself.

Relative neighborhood graph

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 and by an edge whenever there does not exist a third point that is closer to both and 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.

Clique complex

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 graph theory, the De Bruijn–Erdős theorem relates graph coloring of an infinite graph to the same problem on its finite subgraphs. It states that, when all finite subgraphs can be colored with colors, the same is true for the whole graph. The theorem was proved by Nicolaas Govert de Bruijn and Paul Erdős (1951), after whom it is named.

Beta skeleton

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.

Degeneracy (graph theory) Measurement of graph sparsity

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.

In geometry, the moment curve is an algebraic curve in d-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form

In mathematics and probability theory, continuum percolation theory is a branch of mathematics that extends discrete percolation theory to continuous space. More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum percolation are often randomly positioned in some continuous space and form a type of point process. For each point, a random shape is frequently placed on it and the shapes overlap each with other to form clumps or components. As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components. Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study of random graphs and random geometric graphs.

Flip graph

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.

References

  1. 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
  2. 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
  3. Norrenbrock, Christoph (May 2016), "Percolation threshold on planar Euclidean Gabriel graphs", European Physical Journal B , 89 (5), arXiv: 1406.0663 , doi:10.1140/epjb/e2016-60728-0
  4. 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
  5. 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