Circular-arc graph

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.

A circular-arc graph (left) and a corresponding arc model (right).

Formally, let

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

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

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

and

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

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

Recognition

Tucker (1980) demonstrated the first polynomial recognition algorithm for circular-arc graphs, which runs in ${\displaystyle {\mathcal {O}}(n^{3})}$ time. McConnell (2003) gave the first linear ${\displaystyle ({\mathcal {O}}(n+m))}$ time recognition algorithm, where ${\displaystyle m}$ is the number of edges. More recently, Kaplan and Nussbaum^[1] developed a simpler linear time recognition algorithm.

Relation to other graph classes

Circular-arc graphs are a natural generalization of interval graphs. If a circular-arc graph G has an arc model that leaves some point of the circle uncovered, the circle can be cut at that point and stretched to a line, which results in an interval representation. Unlike interval graphs, however, circular-arc graphs are not always perfect, as the odd chordless cycles C₅, C₇, etc., are circular-arc graphs.

Some subclasses

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

Unit circular-arc graphs

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

The number of labelled unit circular-arc graphs on n vertices is given by ${\displaystyle (n+2){\binom {2n-1}{n-1}}-2^{2n-1}}$. ^[2]

Proper circular-arc graphs

${\displaystyle G}$ is a proper circular-arc graph (also known as a circular interval graph)^[3] if there exists a corresponding arc model such that no arc properly contains another. Recognizing these graphs and constructing a proper arc model can both be performed in linear ${\displaystyle ({\mathcal {O}}(n+m))}$ time.^[4] They form one of the fundamental subclasses of the claw-free graphs.^[3]

Helly circular-arc graphs

${\displaystyle G}$ is a Helly circular-arc graph if there exists a corresponding arc model such that the arcs constitute a Helly family. Gavril (1974) gives a characterization of this class that implies an ${\displaystyle {{\mathcal {O}}(n^{3})}}$ recognition algorithm.

Joeris et al. (2009) give other characterizations of this class, which imply a recognition algorithm that runs in O(n+m) time when the input is a graph. If the input graph is not a Helly circular-arc graph, then the algorithm returns a certificate of this fact in the form of a forbidden induced subgraph. They also gave an O(n) time algorithm for determining whether a given circular-arc model has the Helly property.

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.