Circular-arc graph

A circular-arc graph (left) and a set of circular arcs representing it (right).

In graph theory, a circular-arc graph is the intersection graph of a set of arcs on the circle. It has one vertex for each arc in the set, and an edge between every pair of vertices corresponding to arcs that intersect.

Formally, let

$I_{1},I_{2},\ldots ,I_{n}\subset C_{1}$

be a set of arcs. Then the corresponding circular-arc graph is G = (V, E) where

$V=\{I_{1},I_{2},\ldots ,I_{n}\}$

and

$\{I_{\alpha },I_{\beta }\}\in E\iff I_{\alpha }\cap I_{\beta }\neq \varnothing .$

A family of arcs that corresponds to G is called an arc model.

Recognition

Tucker demonstrated the first polynomial recognition algorithm for circular-arc graphs, which runs in ${{\mathcal {O}}(n^{3})}$ time.

More recently, McConnell gave the first linear (${{\mathcal {O}}(n+m)}$) recognition algorithm.

Some subclasses

In the following, let $G=(V,E)$ be an arbitrary graph.

Unit circular-arc graphs

$G$ is a unit circular-arc graph if there exists a corresponding arc model such that each arc is of equal length.

Proper circular-arc graphs

$G$ is a unit circular-arc graph if there exists a corresponding arc model such that no arc properly contains another.

Helly circular-arc graphs

$G$ is a Helly circular-arc graph if there exists a corresponding arc model such that the arcs constitute a Helly family. Gavril gave a forbidden subgraph characterization of this class, which leads to an ${{\mathcal {O}}(n^{3})}$ recognition algorithm.

Applications

Circular-arc graphs are useful in modeling periodic resource allocation problems in operations research. Each interval represents a request for a resource for a specific period repeated in time.

External links

• "circular arc graph". Information System on Graph Class Inclusions. {{cite web}}: External link in |work= (help)