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19
Linear time solvable optimization problems on graphs of bounded cliquewidth
, 2000
"... 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 dec ..."
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Cited by 170 (22 self)
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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.
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 111 (12 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 36 (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.
Finding Maximum Induced Matchings in Subclasses of ClawFree and P5Free Graphs, and in Graphs with Matching and Induced Matching of Equal Maximum Size
, 2003
"... In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. ..."
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Cited by 11 (0 self)
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In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NPhard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NPhard on linegraphs, clawfree graphs, chairfree graphs, Hamiltonian graphs and rregular graphs for r ≥ 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P5, Dm)free graphs, on (bull, chair)free graphs and on linegraphs of Hamiltonian graphs.
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 10 (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.
Computing the Treewidth and the Minimum Fillin With the Modular Decomposition
, 2001
"... Using the notion of modular decomposition we extend the class of graphs on which both the TREEWIDTH and the MINIMUMFILLIN problems can be solved in polynomial time. We show that if C is a class of graphs which is modularly decomposable into graphs that have a polynomial number of minimal separa ..."
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Cited by 9 (2 self)
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Using the notion of modular decomposition we extend the class of graphs on which both the TREEWIDTH and the MINIMUMFILLIN problems can be solved in polynomial time. We show that if C is a class of graphs which is modularly decomposable into graphs that have a polynomial number of minimal separators, or graphs formed by adding a matching between two cliques, then both the TREEWIDTH and the MINIMUMFILLIN problems on C can be solved in polynomial time. For the graphs that are modular decomposable into cycles we give algorithms, that use respectively O(n) and O(n³) time for TREEWIDTH and MINIMUM FILLIN.
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.
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 7 (5 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.