## Linear-Time Recognition of Circular-Arc Graphs (2003)

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Venue: | Algorithmica |

Citations: | 36 - 7 self |

### BibTeX

@MISC{McConnell03linear-timerecognition,

author = {Ross M. McConnell},

title = {Linear-Time Recognition of Circular-Arc Graphs },

year = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

A graph G is a circular-arc graph if it is the intersection graph of a set of arcs on a circle. That is, there is one arc for each vertex of G, and two vertices are adjacent in G if and only if the corresponding arcs intersect. We give a linear-time algorithm for recognizing this class of graphs. When G is a member of the class, the algorithm gives a certificate in the form of a set of arcs that realize it.

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(Show Context)
Citation Context ...e arcs can intersect pairwise around the circle, yet have no point in common. The number of maximal cliques in a circular-arc graph can be exponential in n [27]. It was initially conjectured by Booth =-=[2]-=- that recognition of circular-arc graphs was NP-complete. Tucker disproved this with an O(n 3 ) algorithm [27]. Hsu improved this to O(nm), where m is the number of edges [14], and Eschen and Spinrad ... |

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Citation Context ...intersection graph of a family of n sets is the graph where the vertices are the sets, and the edges are the pairs of sets that intersect. Every graph is the intersection graph of some family of sets =-=[16]-=-. An graph is an interval graph if there is a way to order the universe from which the sets are drawn so that each set is consecutive. Equivalently, a graph is an interval graph if it is the intersect... |

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(Show Context)
Citation Context ...ace of b, and the pivot w is to the left of X. 2 This gives the following algorithm for interval orientation of an interval matrix, which was originally given as a transitive orientation algorithm in =-=[19]-=-, using a lemma analogous to Lemma 6.20 for the transitive orientation problem. Algorithm 6.21 Orient(T)snds an interval orientation of an interval matrix T , using recursive vertex partitioning. Part... |

4 |
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(Show Context)
Citation Context ...he class of circular-arc graphs. Interval graphs and circular-arc graphs arise in scheduling problems and other combinatorial problems. Before the structure of DNA was well-understood, Seymour Benzer =-=[1]-=- was able to show that the set of intersections of a large number of fragments of genetic material in a virus were an interval graph. This provided strong evidence that genetic information was arrange... |

3 |
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(Show Context)
Citation Context ...ng whether N(x) \ N(y) is empty. Theorem 5.3 [26] Neighborhood containments tests between between k pairs of vertices in a chordal bipartite graph can be performed in O(n 1 n 2 + k) time. Theorem 5.4 =-=[7]-=- Disjoint neighborhood tests between between k pairs of vertices in a chordal bipartite graph can be performed in O(n 1 n 2 + k) time. 11 Theorem 5.5 [9]: Let G be an arbitrary circular-arc graph, and... |

2 |
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Citation Context ...n x requires O(jN(x)j) time, giving the bound. This clever idea is due to Spinrad, and initially appeared in an unpublished but in uential manuscript, which he has freely circulated beginning in 1985 =-=[25]-=-. (This paper was also thesrst to propose vertex partitioning as an algorithmic tool, and to recognize its importance to the modular decomposition and transitive orientation problems.) Algorithm 6.24 ... |

1 |
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(Show Context)
Citation Context ...artition on the matrix. Algorithm 6.24 gives an implementation of the while loop of RPartition that allows RPartition to run in O(n+m log n) time. The algorithm is similar to the approach of [19] and =-=[13]-=-. Retrieving E 0 takes time proportional to the sum of degrees of X. The grouping operations on E 0 take O(jE 0 j) time, by distributing the edges to buckets that are initially empty, and maintaining ... |

1 |
Lexicographic breadth-first search: a partition refining technique
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(Show Context)
Citation Context ...tition on the matrix. Algorithm 6.24 gives an implementation of the while loop of RPartition that allows RPartition to run in O(n + m log n) time. The algorithm is similar to the approach of [19] and =-=[13]. Retr-=-ieving E ′ takes time proportional to the sum of degrees of X. The grouping operations on E ′ take O(|E ′ |) time, by distributing the edges to buckets that are initially empty, and maintaining ... |