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Linear time solvable optimization problems on graphs of bounded cliquewidth, Extended abstract
 Graph Theoretic Concepts in Computer Science, 24th International Workshop, WG ’98, Lecture Notes in Computer Science
, 1998
"... Abstract. Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of treewidth at most k, i.e., that have tree decompositions of width at most k, where k is fixed, ..."
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Cited by 113 (20 self)
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Abstract. Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of treewidth at most k, i.e., that have tree decompositions of width at most k, where k is fixed, every decision or optimization problem expressible in monadic secondorder logic has a linear algorithm. We prove that this is also the case for graphs of cliquewidth at most k, where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications. We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with “too many ” induced paths with four vertices. 1.
Modular Decomposition and Transitive Orientation
, 1999
"... A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular ..."
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Cited by 90 (14 self)
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A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n +m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problems on comparability graphs and their complements.
Efficient and practical algorithms for sequential modular decomposition
, 1999
"... A module of an undirected graph G = (V, E) is a set X of vertices that have the same set of neighbors in V \ X. The modular decomposition is a unique decomposition of the vertices into nested modules. We give a practical algorithm with an O(n + m(m;n)) time bound and a variant with a linear time bou ..."
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Cited by 30 (1 self)
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A module of an undirected graph G = (V, E) is a set X of vertices that have the same set of neighbors in V \ X. The modular decomposition is a unique decomposition of the vertices into nested modules. We give a practical algorithm with an O(n + m(m;n)) time bound and a variant with a linear time bound.
Weighted parameters in (P5, P5)free graphs
 Discrete Appl. Math
, 1997
"... We use the modular decomposition to give O(n(m + n)) algorithms for finding a maximum weighted clique (respectively stable set) and an approximate weighted colouring (respectively partition into cliques) in a (P5, P5)free graph. As a byproduct, we obtain an O(m+n) algorithm for finding a minimum w ..."
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Cited by 8 (0 self)
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We use the modular decomposition to give O(n(m + n)) algorithms for finding a maximum weighted clique (respectively stable set) and an approximate weighted colouring (respectively partition into cliques) in a (P5, P5)free graph. As a byproduct, we obtain an O(m+n) algorithm for finding a minimum weighted transversal of the C5 in a (P5, P5) free graph.
An O(n²) Incremental Algorithm for Modular Decomposition of Graphs and 2Structures
 ALGORITHMICA
, 1995
"... This paper gives an O(n²) incremental algorithm for computing the modular decomposition of 2structure [1, 2]. A 2structure is a type of edgecolored graph, and its modular decomposition is also known as the prime tree family. Modular decomposition of 2structures arises in the study of relational ..."
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Cited by 7 (3 self)
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This paper gives an O(n²) incremental algorithm for computing the modular decomposition of 2structure [1, 2]. A 2structure is a type of edgecolored graph, and its modular decomposition is also known as the prime tree family. Modular decomposition of 2structures arises in the study of relational systems. The modular decomposition of undirected graphs and digraphs is a special case, and has applications in a number of combinatorial optimization problems. This algorithm generalizes elements of a previous O(n²) algorithm of Muller and Spinrad [3] for the decomposition of undirected graphs. However, Muller and Spinrad's algorithm employs a sophisticated data structure that impedes its generalization to digraphs and 2structures, and limits its practical use. We replace this data structure with a scheme that labels each edge with at most one node, thereby obtaining an algorithm that is both practical and general to 2structures.
On the CliqueWidth of Graphs with Few P 4 s
, 1998
"... Babel and Olariu (1995) introduced the class of (q; t) graphs in which every set of q vertices has at most t distinct induced P 4 s. Graphs of cliquewidth at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k expressions based on graph operati ..."
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Cited by 3 (1 self)
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Babel and Olariu (1995) introduced the class of (q; t) graphs in which every set of q vertices has at most t distinct induced P 4 s. Graphs of cliquewidth at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k expressions based on graph operations which use k vertex labels. In this paper we study the cliquewidth of the (q; t) graphs, for almost all possible combinations of q and t. On one hand we show that every (q; q \Gamma 3) graph for q 7, has clique width q and a qexpression defining it can be obtained in linear time. On the other hand we show that this result does not hold for the class of (q; q) graphs for any q, and for the class of (q; q \Gamma 3) graphs for q 6. More precisely, we show that for every q, for every n 2 N there is a graph H n which is a (q; q) graph having n vertices and the cliquewidth of H n is at least ( p n=3q)=3q. Partially supported by a Grant of the Israeli Ministry of Science for Fr...
Enumeration of PinPermutations
, 2008
"... In this paper, we study the class of pinpermutations, that is to say of permutations having a pin representation. This class has been recently introduced in [16], where it is used to find properties (algebraicity of the generating function, decidability of membership) of classes of permutations, de ..."
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Cited by 2 (1 self)
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In this paper, we study the class of pinpermutations, that is to say of permutations having a pin representation. This class has been recently introduced in [16], where it is used to find properties (algebraicity of the generating function, decidability of membership) of classes of permutations, depending on the simple permutations this class contains. We give a recursive characterization of the substitution decomposition trees of pinpermutations, which allows us to compute the generating function of this class, and consequently to prove, as it is conjectured in [18], the rationality of this generating function. Moreover, we show that the basis of the pinpermutation class is infinite.
Weighted parameters in (P 5 ,CP 5 )free graphs
"... We use the modular decomposition to give O(n(m + n)) algorithms for finding a maximum weighted clique (respectively stable set) and an approximate weighted colouring (respectively partition into cliques) in a (P5 ; P5 )free graph. As a byproduct, we obtain an O(m+n) algorithm for finding a minimum ..."
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We use the modular decomposition to give O(n(m + n)) algorithms for finding a maximum weighted clique (respectively stable set) and an approximate weighted colouring (respectively partition into cliques) in a (P5 ; P5 )free graph. As a byproduct, we obtain an O(m+n) algorithm for finding a minimum weighted transversal of the C5 in a (P5 ; P5 ) free graph. Keywords: weighted parameters, perfect graphs, modular decomposition 1 Introduction Let G = (V; E) be an arbitrary graph and w a positive integer function defined on V . The classical problems of finding the clique (resp. stability) number and the chromatic (resp. clique cover) number of a (unweighted) graph may be reformulated for a weighted graph denoted (G; w) in the following way: Weighted clique problem. Given a weighted graph (G; w), find a clique C of G such that P x2C w(x) is as large as possible. Weighted colouring problem. Given a weighted graph (G; w), find stable sets S 1 ; S 2 ; : : : ; S t and positive integers...