## Graph Decompositions and Factorizing Permutations (1997)

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Venue: | Discrete Mathematics and Theoretical Computer Science |

Citations: | 9 - 5 self |

### BibTeX

@INPROCEEDINGS{Capelle97graphdecompositions,

author = {Christian Capelle and Michel Habib},

title = {Graph Decompositions and Factorizing Permutations},

booktitle = {Discrete Mathematics and Theoretical Computer Science},

year = {1997},

pages = {132--143},

publisher = {IEEE}

}

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### Abstract

A factorizing permutation of a given undirected graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition and we propose a linear algorithm that computes the whole decomposition tree when a factorizing permutation is provided. This algorithm can be seen as a common generalization of Ma and Hsu [9, 8] for modular decomposition of chordal graphs and Habib, Huchard and Spinrad [7] for inheritance graphs decomposition. It also suggests many new decomposition algorithms for various notions of graph decompositions.

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Citation Context ...the whole decomposition tree when a factorizing permutation is provided. The approach of using parenthesized factors of a list was first used by Hopcroft and Tarjan for triconnected components search =-=[14]-=-. Our algorithm can be seen as a common generalization of Hsu and Ma [16, 15] for modular decomposition of chordal graphs and Habib, Huchard and Spinrad [10] for inheritance graphs decomposition [1]. ... |

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Citation Context ... ⇒ (x,y) ∈ E and (y,x) ∈ E LCA(x,y) is an order node ⇒ either (x,y) ∈ E or (y,x) ∈ E LCA(x,y) is a parallel node ⇒ (x,y) /∈ E and (y,x) /∈ E Although linear decomposition algorithms are now available =-=[4, 20, 7]-=-, they remain rather hard to implement, and it is still worthwhile to search for simplifications. The result presented here can be understood as a step forward in this direction. Moreover, existing al... |

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Citation Context ... ⇒ (x,y) ∈ E and (y,x) ∈ E LCA(x,y) is an order node ⇒ either (x,y) ∈ E or (y,x) ∈ E LCA(x,y) is a parallel node ⇒ (x,y) /∈ E and (y,x) /∈ E Although linear decomposition algorithms are now available =-=[4, 20, 7]-=-, they remain rather hard to implement, and it is still worthwhile to search for simplifications. The result presented here can be understood as a step forward in this direction. Moreover, existing al... |

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Citation Context ...ading example of such a graph decomposition, one can take the modular decomposition, also called substitution decomposition, of graphs (an overview of this theory and its applications can be found in =-=[21]-=-). We present the basic concepts of this decomposition in the following section. In this article we deal with the general case of decomposition of directed graphs. Other examples are modules or blocks... |

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Citation Context ...ently improved down to linear-time [12]. Another direction of research could be to generalize the notion of factorization permutation to other decompositions such as the split decomposition of graphs =-=[6, 18]-=-. This would be very interesting because in such a decomposition the decomposition tree is quite different. Acknowledgements The authors wish to thank the referees for their valuables remarks and sugg... |

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Citation Context ...The approach of using parenthesized factors of a list was first used by Hopcroft and Tarjan for triconnected components search [14]. Our algorithm can be seen as a common generalization of Hsu and Ma =-=[16, 15]-=- for modular decomposition of chordal graphs and Habib, Huchard and Spinrad [10] for inheritance graphs decomposition [1]. It also suggests many new decomposition algorithms for various notions of gra... |

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Citation Context ...The approach of using parenthesized factors of a list was first used by Hopcroft and Tarjan for triconnected components search [14]. Our algorithm can be seen as a common generalization of Hsu and Ma =-=[16, 15]-=- for modular decomposition of chordal graphs and Habib, Huchard and Spinrad [10] for inheritance graphs decomposition [1]. It also suggests many new decomposition algorithms for various notions of gra... |

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11 |
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Citation Context ... Tarjan for triconnected components search [14]. Our algorithm can be seen as a common generalization of Hsu and Ma [16, 15] for modular decomposition of chordal graphs and Habib, Huchard and Spinrad =-=[10]-=- for inheritance graphs decomposition [1]. It also suggests many new decomposition algorithms for various notions of graph decompositions. Keywords: Graph algorithms, graph decompositions, modular dec... |

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Citation Context ...y Ducournau and Habib [8], any depth-first greedy linear extension of an inheritance graph yields a factorizing permutation for the module decomposition of inheritance graphs. Moreover Habib and Paul =-=[12]-=- have proposed a very simple O(n + m) algorithm to compute a factorizing permutation of a cograph. This work has been generalized to modular decomposition [13] with an O(n + mlogn) algorithm for undir... |

5 | Block decomposition of inheritance hierarchies
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Citation Context ... [14]. Our algorithm can be seen as a common generalization of Hsu and Ma [16, 15] for modular decomposition of chordal graphs and Habib, Huchard and Spinrad [10] for inheritance graphs decomposition =-=[1]-=-. It also suggests many new decomposition algorithms for various notions of graph decompositions. Keywords: Graph algorithms, graph decompositions, modular decomposition. 1 General decomposition frame... |

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3 |
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Citation Context ...on. In this article we deal with the general case of decomposition of directed graphs. Other examples are modules or blocks for inheritance acyclic directed graphs leading also to decomposition trees =-=[17, 10, 2, 1]-=-. Clearly it could be hard to find a factorizing permutation, but in some cases a factorizing permutation is given for free. As for example as noticed by Hsu and Ma [16, 15] in the case of chordal gra... |

2 |
An O(n )-Algorithm for Undirected Split Decomposition
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(Show Context)
Citation Context ...ently improved down to linear-time [12]. Another direction of research could be to generalize the notion of factorization permutation to other decompositions such as the split decomposition of graphs =-=[6, 18]-=-. This would be very interesting because in such a decomposition the decomposition tree is quite different. Acknowledgements The authors wish to thank the referees for their valuables remarks and sugg... |

1 |
Partition Refinement techniques: an interesting toolkit
- Habib, Paul, et al.
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(Show Context)
Citation Context ...eritance graphs. Moreover Habib and Paul [12] have proposed a very simple O(n + m) algorithm to compute a factorizing permutation of a cograph. This work has been generalized to modular decomposition =-=[13]-=- with an O(n + mlogn) algorithm for undirected graphs. These algorithms are easy to understand. As in the previous examples, they show the usefulness of dividing the calculation of the decomposition t... |