## Fast and simple algorithms for recognizing chordal comparability graphs and interval graphs (1999)

Venue: | SIAM Journal on Computing |

Citations: | 16 - 3 self |

### BibTeX

@ARTICLE{Hsu99fastand,

author = {Wen-lian Hsu and Tze-heng Ma},

title = {Fast and simple algorithms for recognizing chordal comparability graphs and interval graphs},

journal = {SIAM Journal on Computing},

year = {1999},

volume = {28},

pages = {1004--1020}

}

### OpenURL

### Abstract

Abstract. In this paper, we present a linear-time algorithm for substitution decomposition on chordal graphs. Based on this result, we develop a linear-time algorithm for transitive orientation on chordal comparability graphs, which reduces the complexity of chordal comparability recognition from O(n 2)toO(n+m). We also devise a simple linear-time algorithm for interval graph recognition where no complicated data structure is involved. Key words. chordal graph, triangulated graph, interval graph, analysis of algorithms, graph theory, substitution decomposition, modular decomposition, cycle-free poset, transitive orientation, graph partitioning, cardinality lexicographic ordering, graph recognition

### Citations

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(Show Context)
Citation Context ...ate the orientation algorithm with the substitution decomposition tree for a chordal graph. It is well known that the orientation within a module is independent to the orientation outside that module =-=[9]-=-. Formally, a graph is comparability iff each of its modules (including the prime graph represented at the root of a decomposition tree) can be transitively oriented. Since each module of a chordal gr... |

798 |
Matrix multiplication via arithmetic progressions
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(Show Context)
Citation Context ... test whether this directed graph is transitive. The fastest algorithm for the latter problem takes time proportional to that of multiplying two n × n Boolean matrices, which is currently O(n 2.376 ) =-=[3]-=-. Recently, an O(n + m) algorithm has been developed to test whether a directed chordal graph is transitives1006 WEN-LIAN HSU AND TZE-HENG MA [16]. This brings the complexity of recognizing chordal co... |

468 |
Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms
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- 1976
(Show Context)
Citation Context ...ent iff their corresponding intervals overlap. Interval graphs are exactly the chordal co-comparability graphs [10]. This class of graphs has a wide range of applications (cf. [11]). Booth and Lueker =-=[1]-=- devised the first linear-time algorithm to recognize interval graphs using a complicated data structure called a PQ-tree. Korte and Möhring [11] simplified the operations on a PQ-tree by carrying out... |

293 |
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(Show Context)
Citation Context ...of v that are ordered after v form a clique. A perfect elimination scheme of a chordal graph can be obtained by taking the reverse of a lexicographic ordering, which can be carried out in linear time =-=[18]-=-. A lexicographic ordering can be considered a special kind of breadth-first ordering. Imagine there is a label of n digits, initially filled with zeros, associated with each vertex. After the ith ver... |

191 |
Incidence matrices and interval graphs
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(Show Context)
Citation Context ...have been a very useful model for many applications. Interested readers are referred to [6], [9]. The fastest algorithm to recognize interval graphs relies on the following property. Theorem 5.1 (See =-=[7]-=-). A graph G is an interval graph iff its maximal cliques can be linearly ordered such that, for each vertex v, the maximal cliques containing v occur consecutively. This linear ordering of the maxima... |

139 |
Interval orders and interval graphs
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(Show Context)
Citation Context ...Chordal comparability graphs can be recognized in O(m + n) time. 5. Interval graph recognition. Interval graphs have been a very useful model for many applications. Interested readers are referred to =-=[6]-=-, [9]. The fastest algorithm to recognize interval graphs relies on the following property. Theorem 5.1 (See [7]). A graph G is an interval graph iff its maximal cliques can be linearly ordered such t... |

138 |
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(Show Context)
Citation Context ...d for partitioning. This contradicts v ′ ∈ S. Since G is prime, S cannot exist and every nonsimplicial vertex must be marked by the algorithm. The O(n + m) complexity comes from an amortized analysis =-=[22]-=-. The clique tree and the intersection of all pairs of adjacent maximal cliques can be obtained in linear time. When we traverse the clique tree, whenever an edge is passed, every vertex recorded in t... |

133 |
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(Show Context)
Citation Context ... yields an O(n + m) algorithm for chordal comparability graph recognition. A graph G =(V,E) is called an interval graph if it is the intersection graph of a set F of closed intervals on the real line =-=[14]-=-. In other words, there is a one-toone mapping between the vertices in G and the intervals in F such that two vertices are adjacent iff their corresponding intervals overlap. Interval graphs are exact... |

126 |
The intersection graphs of subtrees in trees are exactly the chordal graphs
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(Show Context)
Citation Context ...al graphs is that the maximal cliques of a chordal graph can be connected to form a tree T such that for each vertex v, the subgraph induced on T by the maximal cliques containing v is connected [2], =-=[8]-=-. (Call it the connectivity property.) Our algorithm applies a partitioning technique on the clique tree structure for the input chordal graph. At the end, all nonsimplicial vertices are marked either... |

84 |
A characterization of rigid circuit graphs
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(Show Context)
Citation Context ...chordal graphs is that the maximal cliques of a chordal graph can be connected to form a tree T such that for each vertex v, the subgraph induced on T by the maximal cliques containing v is connected =-=[2]-=-, [8]. (Call it the connectivity property.) Our algorithm applies a partitioning technique on the clique tree structure for the input chordal graph. At the end, all nonsimplicial vertices are marked e... |

84 |
A characterization of comparability graphs and of interval graphs
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(Show Context)
Citation Context ...mapping between the vertices in G and the intervals in F such that two vertices are adjacent iff their corresponding intervals overlap. Interval graphs are exactly the chordal co-comparability graphs =-=[10]-=-. This class of graphs has a wide range of applications (cf. [11]). Booth and Lueker [1] devised the first linear-time algorithm to recognize interval graphs using a complicated data structure called ... |

62 |
Linear-time modular decomposition and efficient transitive orientation of comparability graphs
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- 1994
(Show Context)
Citation Context ...s proved the following corollary. Corollary 5.5. An interval graph can be recognized in linear time.s1020 WEN-LIAN HSU AND TZE-HENG MA Remark. After we submitted our paper [12], McConnell and Spinrad =-=[17]-=- have developed a linear-time algorithm to perform modular decomposition for general graphs. They also discussed the transitive orientation on a number of problems. Their time bound for recognizing ch... |

54 |
An incremental linear-time algorithm for recognizing interval graphs
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(Show Context)
Citation Context ...t two vertices are adjacent iff their corresponding intervals overlap. Interval graphs are exactly the chordal co-comparability graphs [10]. This class of graphs has a wide range of applications (cf. =-=[11]-=-). Booth and Lueker [1] devised the first linear-time algorithm to recognize interval graphs using a complicated data structure called a PQ-tree. Korte and Möhring [11] simplified the operations on a ... |

51 |
Bipartite permutation graphs
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(Show Context)
Citation Context ...ue linear maximal clique arrangement up to the reversal of the interval model). A chordal comparability graph is a graph which is both a chordal graph and a comparability graph. The fastest algorithm =-=[20]-=- for recognizing a comparability graph involves two stages. First, the input graph is transitively oriented, which can be done in O(n 2 ) time. Then we test whether this directed graph is transitive. ... |

38 |
Substitution decomposition on chordal graphs and applications
- Hsu, Ma
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(Show Context)
Citation Context ... interval graph. We have thus proved the following corollary. Corollary 5.5. An interval graph can be recognized in linear time.s1020 WEN-LIAN HSU AND TZE-HENG MA Remark. After we submitted our paper =-=[12]-=-, McConnell and Spinrad [17] have developed a linear-time algorithm to perform modular decomposition for general graphs. They also discussed the transitive orientation on a number of problems. Their t... |

35 | The ultimate interval graph recognition algorithm? (extended abstract
- Corneil, Olariu, et al.
- 1998
(Show Context)
Citation Context ...bound for recognizing chordal comparability graphs is O(n + mlogn). Yet another linear-time algorithm for recognizing interval graph without using PQ-tree is developed by Corneil, Olariu, and Stewart =-=[4]-=-. Acknowledgment. The authors thank the referees who made many helpful suggestions and a clearer presentation for procedure Orientation and the paper. We also want to thank Spinrad who pointed out an ... |

25 |
Incremental modular decomposition
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(Show Context)
Citation Context ...ree represents a nontrivial module marked by its root. An example of a substitution decomposition is shown in Fig. 1.1. For general graphs, substitution decomposition takes O(min(n 2 ,mα(m, n))) time =-=[15]-=-, [21]. In this paper, we call a graph prime if it does not contain a nontrivial module. ∗Received by the editors January 17, 1992; accepted for publication February 13, 1998; published electronically... |

23 |
O(M · N) algorithms for the recognition and isomorphism problems on circular-arc graphs
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- 1995
(Show Context)
Citation Context ...lity graph, there is a unique transitive orientation [19] (up to the reversal of the directions of all edges); (ii) if it is an interval graph, there is a unique interval representation for the graph =-=[13]-=- (by which we mean there is a unique linear maximal clique arrangement up to the reversal of the interval model). A chordal comparability graph is a graph which is both a chordal graph and a comparabi... |

20 |
P4-trees and substitution decomposition
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(Show Context)
Citation Context ...presents a nontrivial module marked by its root. An example of a substitution decomposition is shown in Fig. 1.1. For general graphs, substitution decomposition takes O(min(n 2 ,mα(m, n))) time [15], =-=[21]-=-. In this paper, we call a graph prime if it does not contain a nontrivial module. ∗Received by the editors January 17, 1992; accepted for publication February 13, 1998; published electronically Janua... |

13 | Linear-time modular decomposition and ecient transitive orientation of comparability graphs - McConnell, Spinrad - 1994 |

7 |
Minimizing setups for cycle-free orders sets
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(Show Context)
Citation Context ...ertex set V , R as the edge set E. Chordal comparability graphs, when transitively oriented, become a class of poset called cycle-free posets. For more about characteristics on cycle-free posets, see =-=[5]-=-, [16]. A poset can be expressed by its Hasse diagram, which is an undirected graph with a minimum number of edges where there is an upward path from a to b iff a dominates b. Achain of a poset is a p... |

5 |
Cycle-free partial orders and chordal comparability graphs, Order 8
- Ma, Spinrad
- 1991
(Show Context)
Citation Context ...n × n Boolean matrices, which is currently O(n 2.376 ) [3]. Recently, an O(n + m) algorithm has been developed to test whether a directed chordal graph is transitives1006 WEN-LIAN HSU AND TZE-HENG MA =-=[16]-=-. This brings the complexity of recognizing chordal comparability graphs down to O(n 2 ). The new bottleneck is the transitive orientation of chordal graphs. In section 4, we present an algorithm whic... |

5 |
Partially ordered sets and their comparability graphs
- Shevrin, Filippov
- 1970
(Show Context)
Citation Context ...this ordering. This algorithm decomposes chordal graphs into prime components. A prime graph has the following properties: (i) if it is a comparability graph, there is a unique transitive orientation =-=[19]-=- (up to the reversal of the directions of all edges); (ii) if it is an interval graph, there is a unique interval representation for the graph [13] (by which we mean there is a unique linear maximal c... |