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Concept
Unit disk graph
Summary
In geometric graph theory, a unit disk graph is the intersection graph of a family of unit disks in the Euclidean plane. That is, it is a graph with one vertex for each disk in the family, and with an edge between two vertices whenever the corresponding vertices lie within a unit distance of each other.
They are commonly formed from a Poisson point process, making them a simple example of a random structure.
There are several possible definitions of the unit disk graph, equivalent to each other up to a choice of scale factor:
Unit disk graphs are the graph formed from a collection of points in the Euclidean plane, with a vertex for each point and an edge connecting each pair of points whose distance is below a fixed threshold.
Unit disk graphs are the intersection graphs of equal-radius circles, or of equal-radius disks. These graphs have a vertex for each circle or disk, and an edge connecting each pair of circles or disks that have a nonempty intersection.
Unit disk graphs may be formed in a different way from a collection of equal-radius circles, by connecting two circles with an edge whenever one circle contains the center of the other circle.
Every induced subgraph of a unit disk graph is also a unit disk graph. An example of a graph that is not a unit disk graph is the star with one central node connected to six leaves: if each of six unit disks touches a common unit disk, some two of the six disks must touch each other. Therefore, unit disk graphs cannot contain an induced subgraph. Infinitely many other forbidden induced subgraphs are known.
The number of unit disk graphs on labeled vertices is within an exponential factor of . This rapid growth implies that unit disk graphs do not have bounded twin-width.
Beginning with the work of , unit disk graphs have been used in computer science to model the topology of ad hoc wireless communication networks. In this application, nodes are connected through a direct wireless connection without a base station. It is assumed that all nodes are homogeneous and equipped with omnidirectional antennas.
Official source
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