scholarly journals Graph Decompositions and Factorizing Permutations

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Christian Capelle ◽  
Michel Habib ◽  
Fabien Montgolfier
Keyword(s):  
Graph Decomposition ◽  
Chordal Graphs ◽  
Decomposition Algorithms ◽  
Decomposition Tree ◽  
Modular Decomposition ◽  
Graph Decompositions ◽  
Linear Algorithm ◽  
Common Generalization ◽  
International Audience

International audience A factorizing permutation of a given 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 for modular decomposition of chordal graphs and Habib, Huchard and Spinrad for inheritance graphs decomposition. It also suggests many new decomposition algorithms for various notions of graph decompositions.

scholarly journals A linear algorithm for the group path problem on chordal graphs

1993 ◽  
Vol 44 (1-3) ◽  
pp. 185-190 ◽  
Author(s):  
Srinivasa R. Arikati ◽  
Uri N. Peled
Keyword(s):  
Chordal Graphs ◽  
Linear Algorithm

scholarly journals An Edge-Signed Generalization of Chordal Graphs, Free Multiplicities on Braid Arrangements, and Their Characterizations

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Takuro Abe ◽  
Koji Nuida ◽  
Yasuhide Numata
Keyword(s):  
Special Kind ◽  
Chordal Graph ◽  
Hyperplane Arrangements ◽  
Chordal Graphs ◽  
Nous Proposons ◽  
Signed Graphs ◽  
International Audience ◽  
Perfect Elimination Ordering

International audience In this article, we propose a generalization of the notion of chordal graphs to signed graphs, which is based on the existence of a perfect elimination ordering for a chordal graph. We give a special kind of filtrations of the generalized chordal graphs, and show a characterization of those graphs. Moreover, we also describe a relation between signed graphs and a certain class of multiarrangements of hyperplanes, and show a characterization of free multiarrangements in that class in terms of the generalized chordal graphs, which generalizes a well-known result by Stanley on free hyperplane arrangements. Finally, we give a remark on a relation of our results with a recent conjecture by Athanasiadis on freeness characterization for another class of hyperplane arrangements. Dans cet article, nous proposons une généralisation de la notion des graphes triangulés à graphes signés, qui est basée sur l'existence d'un ordre d'élimination simplicial à un graphe triangulé. Nous donnons un genre spécial de filtrations des graphes triangulés généralisés, et montrons une caractérisation de ces graphes. De plus, nous décrivons aussi une relation entre graphes signés et une certaine classe de multicompositions d'hyperplans, et montrons une caractérisation de multicompositions libres dans cette classe en termes des graphes triangulés généralisés, qui généralise un résultat célèbre de Stanley sur compositions libres d'hyperplans. Finalement, nous donnons une remarque sur une relation de nos résultats avec une conjecture récente d'Athanasiadis sur une caractérisation du freeness d'une autre classe de compositions d'hyperplans.

scholarly journals On P_4-tidy graphs

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
V. Giakoumakis ◽  
F. Roussel ◽  
H. Thuillier
Keyword(s):  
Optimization Problems ◽  
Linear Time ◽  
Hamiltonian Path ◽  
Clique Number ◽  
Stability Number ◽  
Sparse Graphs ◽  
Modular Decomposition ◽  
New Class ◽  
International Audience ◽  
Linear Algorithms

International audience We study the P_4-tidy graphs, a new class defined by Rusu [30] in order to illustrate the notion of P_4-domination in perfect graphs. This class strictly contains the P_4-extendible graphs and the P_4-lite graphs defined by Jamison & Olariu in [19] and [23] and we show that the P_4-tidy graphs and P_4-lite graphs are closely related. Note that the class of P_4-lite graphs is a class of brittle graphs strictly containing the P_4-sparse graphs defined by Hoang in [14]. McConnel & Spinrad [2] and independently Cournier & Habib [5] have shown that the modular decomposition tree of any graph is computable in linear time. For recognizing in linear time P_4-tidy graphs, we apply a method introduced by Giakoumakis in [9] and Giakoumakis & Fouquet in [6] using modular decomposition of graphs and we propose linear algorithms for optimization problems on such graphs, as clique number, stability number, chromatic number and scattering number. We show that the Hamiltonian Path Problem is linear for this class of graphs. Our study unifies and generalizes previous results of Jamison & Olariu ([18], [21], [22]), Hochstattler & Schindler[16], Jung [25] and Hochstattler & Tinhofer [15].

scholarly journals On economical set representations of graphs

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Jing Kong ◽  
Yaokun Wu
Keyword(s):  
Chordal Graphs ◽  
Minimum Size ◽  
Extremal Graphs ◽  
Intersection Numbers ◽  
International Audience ◽  
Perfect Elimination Ordering ◽  
Set Representations

Graphs and Algorithms International audience In this paper we discuss the bounds of and relations among various kinds of intersection numbers of graphs. Especially, we address extremal graphs with respect to the established bounds. The uniqueness of the minimum-size intersection representations for some graphs is also studied. In the course of this work, we introduce a superclass of chordal graphs, defined in terms of a generalization of simplicial vertex and perfect elimination ordering.

scholarly journals Ordered Vertex Partitioning

2000 ◽  
Vol Vol. 4 no. 1 ◽  
Author(s):  
Ross M. Mcconnell ◽  
Jeremy P. Spinrad
Keyword(s):  
Canonical Representation ◽  
Modular Decomposition ◽  
Running Time ◽  
Dual Problems ◽  
International Audience ◽  
Transitive Orientation ◽  
Time Bounds

International audience A transitive orientation of a graph is an orientation of the edges that produces a transitive digraph. The modular decomposition of a graph is a canonical representation of all of its modules. Finding a transitive orientation and finding the modular decomposition are in some sense dual problems. In this paper, we describe a simple O(n + m \log n) algorithm that uses this duality to find both a transitive orientation and the modular decomposition. Though the running time is not optimal, this algorithm is much simpler than any previous algorithms that are not Ω (n^2). The best known time bounds for the problems are O(n+m) but they involve sophisticated techniques.

Drawing Clustered Graph Using Modular Decomposition Tree

2013 ◽  
Vol 12 (8) ◽  
pp. 1491-1501
Author(s):  
Wan-Yu Deng ◽  
Kai Zhang ◽  
Qing-Hua Zheng ◽  
Wei Wei
Keyword(s):  
Decomposition Tree ◽  
Modular Decomposition
Sign in / Sign up

Export Citation Format

Share Document