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Circular-arc graph
Intersection graph for a set of arcs on a circle

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 , … , I n ⊂ C 1 {\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

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

and

{ I α , I β } ∈ E ⟺ I α ∩ I β ≠ ∅ . {\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.

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Recognition

Tucker (1980) demonstrated the first polynomial recognition algorithm for circular-arc graphs, which runs in O ( n 3 ) {\displaystyle {\mathcal {O}}(n^{3})} time. McConnell (2003) gave the first linear ( O ( n + m ) ) {\displaystyle ({\mathcal {O}}(n+m))} time recognition algorithm, where m {\displaystyle m} is the number of edges. More recently, Kaplan and Nussbaum1 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 G = ( V , E ) {\displaystyle G=(V,E)} be an arbitrary graph.

Unit circular-arc graphs

G {\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 ( n + 2 ) ( 2 n − 1 n − 1 ) − 2 2 n − 1 {\displaystyle (n+2){\binom {2n-1}{n-1}}-2^{2n-1}} . 2

Proper circular-arc graphs

G {\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 ( O ( n + m ) ) {\displaystyle ({\mathcal {O}}(n+m))} time.4 They form one of the fundamental subclasses of the claw-free graphs.5

Helly circular-arc graphs

G {\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 O ( n 3 ) {\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.

Notes

References

  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. /wiki/CiteSeerX_(identifier) ↩

  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. /wiki/Greta_Panova ↩

  3. Described with a different but equivalent definition by Chudnovsky & Seymour (2008). - Chudnovsky, Maria; Seymour, Paul (2008), "Claw-free graphs. III. Circular interval graphs" (PDF), Journal of Combinatorial Theory, Series B, 98 (4): 812–834, doi:10.1016/j.jctb.2008.03.001, MR 2418774 http://www.columbia.edu/~mc2775/claws3.pdf ↩

  4. Deng, Hell & Huang (1996) pg. ? - Deng, Xiaotie; Hell, Pavol; Huang, Jing (1996), "Linear-Time representation algorithms for proper circular-arc graphs and proper interval graphs", SIAM Journal on Computing, 25 (2): 390–403, doi:10.1137/S0097539792269095 https://doi.org/10.1137%2FS0097539792269095 ↩

  5. Described with a different but equivalent definition by Chudnovsky & Seymour (2008). - Chudnovsky, Maria; Seymour, Paul (2008), "Claw-free graphs. III. Circular interval graphs" (PDF), Journal of Combinatorial Theory, Series B, 98 (4): 812–834, doi:10.1016/j.jctb.2008.03.001, MR 2418774 http://www.columbia.edu/~mc2775/claws3.pdf ↩