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.
be a set of arcs. Then the corresponding circular-arc graph is G = (V, E) where
A family of arcs that corresponds to G is called an arc model.
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. More recently, Kaplan and Nussbaum 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.
In the following, let be an arbitrary graph.
Unit circular-arc graphs
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
is a proper circular-arc graph (also known as a circular interval graph) 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. They form one of the fundamental subclasses of the claw-free graphs.
Helly circular-arc graphs
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.
- 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.
- 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.
- Gavril, Fanica (1974), "Algorithms on circular-arc graphs", Networks, 4 (4): 357–369, doi:10.1002/net.3230040407.
- Golumbic, Martin Charles (1980), Algorithmic Graph Theory and Perfect Graphs, Academic Press, ISBN 0-444-51530-5. Second edition, Annals of Discrete Mathematics 57, Elsevier, 2004.
- Joeris, Benson L.; Lin, Min Chih; McConnell, Ross M.; Spinrad, Jeremy P.; Szwarcfiter, Jayme L. (2009), "Linear-Time Recognition of Helly Circular-Arc Models and Graphs", Algorithmica, 59 (2): 215–239, doi:10.1007/s00453-009-9304-5.
- McConnell, Ross (2003), "Linear-time recognition of circular-arc graphs", Algorithmica, 37 (2): 93–147, doi:10.1007/s00453-003-1032-7.
- Tucker, Alan (1980), "An efficient test for circular-arc graphs", SIAM Journal on Computing, 9 (1): 1–24, doi:10.1137/0209001.