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Computing common intervals of K permutations, with applications to modular decomposition of graphs
, 2008
"... We introduce a new approach to compute the common intervals of K permutations based on a very simple and general notion of generators of common intervals. This formalism leads to simple and efficient algorithms to compute the set of all common intervals of K permutations, that can contain a quadrat ..."
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Cited by 35 (15 self)
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We introduce a new approach to compute the common intervals of K permutations based on a very simple and general notion of generators of common intervals. This formalism leads to simple and efficient algorithms to compute the set of all common intervals of K permutations, that can contain a quadratic number of intervals, as well as a linear space basis of this set of common intervals. Finally, we show how our results on permutations can be used for computing the modular decomposition of graphs.
Revisiting T. Uno and M. Yagiura’s Algorithm
, 2005
"... ... of two given permutations of length n in O(n + K) time. Our paper first presents a decomposition approach to obtain a compact encoding for common intervals of d permutations. Then, we revisit T. Uno and M. Yagiura’s algorithm to yield a linear time algorithm for finding this encoding. Besides, w ..."
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Cited by 23 (7 self)
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... of two given permutations of length n in O(n + K) time. Our paper first presents a decomposition approach to obtain a compact encoding for common intervals of d permutations. Then, we revisit T. Uno and M. Yagiura’s algorithm to yield a linear time algorithm for finding this encoding. Besides, we adapt the algorithm to obtain a linear time modular decomposition of an undirected graph, and thereby propose a formal invariantbased proof for all these algorithms.
A simple lineartime modular decomposition algorithm for graphs, using order extension
, 2004
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A simple linear time algorithm for cograph recognition
 Discrete Applied Mathematics
, 2005
"... www.elsevier.com/locate/dam ..."
Homogeneity vs. adjacency: generalising some graph decomposition algorithms
 In 32nd International Workshop on GraphTheoretic Concepts in Computer Science (WG), volume 4271 of LNCS
, 2006
"... Abstract. In this paper, a new general decomposition theory inspired from modular graph decomposition is presented. Our main result shows that, within this general theory, most of the nice algorithmic tools developed for modular decomposition are still efficient. This theory not only unifies the usu ..."
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Cited by 6 (3 self)
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Abstract. In this paper, a new general decomposition theory inspired from modular graph decomposition is presented. Our main result shows that, within this general theory, most of the nice algorithmic tools developed for modular decomposition are still efficient. This theory not only unifies the usual modular decomposition generalisations such as modular decomposition of directed graphs and of 2structures, but also decomposition by star cutsets. 1
Simpler LinearTime Modular Decomposition Via Recursive Factorizing Permutations
 in: Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science 5125
"... Abstract. Modular decomposition is fundamental for many important problems in algorithmic graph theory including transitive orientation, the recognition of several classes of graphs, and certain combinatorial optimization problems. Accordingly, there has been a drive towards a practical, lineartime ..."
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Cited by 5 (0 self)
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Abstract. Modular decomposition is fundamental for many important problems in algorithmic graph theory including transitive orientation, the recognition of several classes of graphs, and certain combinatorial optimization problems. Accordingly, there has been a drive towards a practical, lineartime algorithm for the problem. This paper posits such an algorithm; we present a lineartime modular decomposition algorithm that proceeds in four straightforward steps. This is achieved by introducing the notion of factorizing permutations to an earlier recursive approach. The only data structure used is an ordered list of trees, and each of the four steps amounts to simple traversals of these trees. Previous algorithms were either exceedingly complicated or resorted to impractical datastructures. 1
Fullydynamic recognition algorithm and certificate for directed cographs
 Proc. 30th Int’l Workshop on GraphTheoretic Concepts in Computer Science (WG’04), LNCS3353
, 2004
"... Abstract. This paper presents an optimal fullydynamic recognition algorithm for directed cographs. Given the modular decomposition tree of a directed cograph G, the algorithm supports arc and vertex modification (insertion or deletion) in O(d) timewhered is the number of arcs involved in the operat ..."
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Cited by 3 (2 self)
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Abstract. This paper presents an optimal fullydynamic recognition algorithm for directed cographs. Given the modular decomposition tree of a directed cograph G, the algorithm supports arc and vertex modification (insertion or deletion) in O(d) timewhered is the number of arcs involved in the operation. Moreover, if the modified graph remains a directed cograph, the modular tree decomposition is updated; otherwise, a certificate is returned within the same complexity. 1
Simple, Lineartime Modular Decomposition (Extended Abstract)
, 2008
"... Modular decomposition is fundamental for many important problems in algorithmic graph theory including transitive orientation, the recognition of several classes of graphs, and certain combinatorial optimization problems. Accordingly, there has been a drive towards a practical, lineartime algorithm ..."
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Modular decomposition is fundamental for many important problems in algorithmic graph theory including transitive orientation, the recognition of several classes of graphs, and certain combinatorial optimization problems. Accordingly, there has been a drive towards a practical, lineartime algorithm for the problem. Despite considerable effort, such an algorithm has remained elusive. The lineartime algorithms to date are impractical and of mainly theoretical interest. In this paper we present the first simple, lineartime algorithm to compute the modular decomposition tree of an undirected graph.
Abstract ARTICLE IN PRESS Discrete Applied Mathematics ( ) –
, 2006
"... www.elsevier.com/locate/dam This paper presents an optimal fully dynamic recognition algorithm for directed cographs. Given the modular decomposition tree of a directed cograph G, the algorithm supports arc and vertex modification (insertion or deletion) in O(d) time where d is the number of arcs in ..."
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www.elsevier.com/locate/dam This paper presents an optimal fully dynamic recognition algorithm for directed cographs. Given the modular decomposition tree of a directed cograph G, the algorithm supports arc and vertex modification (insertion or deletion) in O(d) time where d is the number of arcs involved in the operation. Moreover, if the modified graph remains a directed cograph, the modular decomposition tree is updated; otherwise, a certificate is returned within the same complexity. © 2006 Elsevier B.V. All rights reserved.
B.M. BuiXuan 1 and M. Habib 2
"... Abstract. We give a quadratic O(X  2) space representation based on a canonical tree for any subset family F ⊆ 2 X closed under the union and the difference of its overlapping members. The cardinality of F is potentially in O(2 X ), and the total cardinality of its members even higher. As far as ..."
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Abstract. We give a quadratic O(X  2) space representation based on a canonical tree for any subset family F ⊆ 2 X closed under the union and the difference of its overlapping members. The cardinality of F is potentially in O(2 X ), and the total cardinality of its members even higher. As far as we know this is the first representation result for such families. As an application of this framework we obtain a unique digraph decomposition that not only captures, but also is strictly more powerful than the wellstudied modular decomposition. A polynomial time decomposition algorithm for this case is described. 1