Unit disk graph

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:

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 K_1,7 with one central node connected to seven leaves: if each of seven unit disks touches a common unit disk, some two of the seven disks must touch each other (as the kissing number in the plane is 6). Therefore, unit disk graphs cannot contain an induced K_1,7 subgraph.

Beginning with the work of Huson & Sen (1995), 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. Node locations are modelled as Euclidean points, and the area within which a signal from one node can be received by another node is modelled as a circle. If all nodes have transmitters of equal power, these circles are all equal. Random geometric graphs, formed as unit disk graphs with randomly generated disk centres, have also been used as a model of percolation and various other phenomena.

If one is given a collection of unit disks (or their centres) in a space of any fixed dimension, it is possible to construct the corresponding unit disk graph in linear time, by rounding the centres to nearby integer grid points, using a hash table to find all pairs of centres within constant distance of each other, and filtering the resulting list of pairs for the ones whose circles intersect. The ratio of the number of pairs considered by this algorithm to the number of edges in the eventual graph is a constant, giving the linear time bound. However, this constant grows exponentially as a function of the dimension (Bentley, Stanat & Williams 1977).

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