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

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

and

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

Recognition

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Tucker (1980) demonstrated the first polynomial recognition algorithm for circular-arc graphs, which runs in   time. McConnell (2003) gave the first linear   time recognition algorithm, where   is the number of edges. More recently, Kaplan and Nussbaum[1] developed a simpler linear time recognition algorithm.

Relation to other graph classes

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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

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In the following, let   be an arbitrary graph.

Unit circular-arc graphs

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  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  . [2]

Proper circular-arc graphs

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  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   time.[4] They form one of the fundamental subclasses of the claw-free graphs.[3]

Helly circular-arc graphs

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  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   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

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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.

Notes

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  1. ^ Kaplan, Haim; Nussbaum, Yahav (2011-11-01). "A Simpler Linear-Time Recognition of Circular-Arc Graphs". Algorithmica. 61 (3): 694–737. CiteSeerX 10.1.1.76.2480. doi:10.1007/s00453-010-9432-y. ISSN 0178-4617.
  2. ^ Alexandersson, Per; Panova, Greta (December 2018). "LLT polynomials, chromatic quasisymmetric functions and graphs with cycles". Discrete Mathematics. 341 (12): 3453–3482. arXiv:1705.10353. doi:10.1016/j.disc.2018.09.001.
  3. ^ a b Described with a different but equivalent definition by Chudnovsky & Seymour (2008).
  4. ^ Deng, Hell & Huang (1996) pg. ?

References

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