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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.
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.
Finiteness Theorems for Graphs and Posets Obtained by Compositions
 Order
, 1998
"... We investigate classes of graphs and posets that admit decompositions to obtain or disprove finiteness results for obstruction sets. To do so we develop a theory of minimal infinite antichains that allows us to characterize such antichains by means of the set of elements below it. In particular we ..."
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Cited by 4 (0 self)
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We investigate classes of graphs and posets that admit decompositions to obtain or disprove finiteness results for obstruction sets. To do so we develop a theory of minimal infinite antichains that allows us to characterize such antichains by means of the set of elements below it. In particular we show that the following classes have infinite antichains with resp. to the induced subgraph/poset relation: interval graphs and orders, Nfree orders, orders with bounded decomposition width. On the other hand for orders with bounded decomposition diameter finiteness of all antichains is shown. As a consequence those classes with infinite antichains have undecidable hereditary properties whereas those with finite antichains have fast algorithms for all such properties.
Cograph Recognition Algorithm Revisited and Online Induced P 4 Search
, 1994
"... . In 1985, Corneil, Perl and Stewart [CPS85] gave a linear incremental algorithm to recognize cographs (graphs with no induced P4 ). When this algorithm stops, either the initial graph is a cograph and the cotree of the whole graph has been built, or the initial graph is not a cograph and this algo ..."
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Cited by 1 (0 self)
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. In 1985, Corneil, Perl and Stewart [CPS85] gave a linear incremental algorithm to recognize cographs (graphs with no induced P4 ). When this algorithm stops, either the initial graph is a cograph and the cotree of the whole graph has been built, or the initial graph is not a cograph and this algorithm ends up with a vertex v and a cotree cot such that v cannot be inserted in cot; so the input graph must contain a P4 . In many applications such as graph decomposition [Cou93, CH93a, CH93b, CH94, EGMS94, Spi92, MS94], transitive orientation [Spi83, ST94], not only the existence but a P4 is also explicitly needed. In this paper, we present a new characterization of cograph in terms of its modular structure. This characterization yields a structural labeling of the cotree for incremental cograph recognition, and we show how to go from this labeling to the Corneil et al. one's. Furthermore, we show how to adapt this algorithm in order to produce a P4 in case of failure when adding a new v...
Nesting of Prime Substructures in kary Relations
, 1999
"... The modular decomposition of a graph or relation has a large number of combinatorial applications. It divides the structure into a set of "prime" induced substructures, which cannot be further decomposed. Recent work on graphs and kary relations has focused on the discovery that prime induced su ..."
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The modular decomposition of a graph or relation has a large number of combinatorial applications. It divides the structure into a set of "prime" induced substructures, which cannot be further decomposed. Recent work on graphs and kary relations has focused on the discovery that prime induced substructures are densely nested when they occur. Lower bounds on the "nesting density" of prime substructures in graphs are used heavily in the only known lineartime algorithm for directed graphs. We improve on the previously known lower bounds for kary relations, and show that no further improvement is possible. 1 Introduction Given a finite undirected graph G = (V; E), let V (G) denote V , and let n(G) denote jV (G)j, let GjX denote the subgraph of G induced by X, let G \Gamma X denote Gj(V (G) \Gamma X), and G \Gamma x denote G \Gamma fxg. Two sets X and Y overlap if X " Y , X \Gamma Y , and Y \Gamma X are all nonempty. A module of G is a set X of nodes such that for any node x no...
The C3structure of the tournaments
, 2005
"... Let T = (V, E) be a tournament. The C3structure of T is the family C3(T) of the subsets {x, y, z} of V such that the subtournament T ({x, y, z}) is a cycle on 3 vertices. In another respect, a subset X of V is an interval of T provided that for a, b ∈ X and x ∈ V − X, (a, x) ∈ E if and only if (b, ..."
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Let T = (V, E) be a tournament. The C3structure of T is the family C3(T) of the subsets {x, y, z} of V such that the subtournament T ({x, y, z}) is a cycle on 3 vertices. In another respect, a subset X of V is an interval of T provided that for a, b ∈ X and x ∈ V − X, (a, x) ∈ E if and only if (b, x) ∈ E. For example, ∅, {x}, where x ∈ V, and V are intervals of T, called trivial intervals. A tournament is indecomposable if all its intervals are trivial. Lastly, with each tournament T = (V, E) is associated the dual tournament T ⋆ = (V, E ⋆ ) defined as: for x, y ∈ V, (x, y) ∈ E ⋆ if (y, x) ∈ E. The following theorem is proved. Given tournaments T = (V, E) and T = (V, E ′ ) such that C3(T) = C3(T ′), if T is indecomposable, then T ′ = T or T ′ = T ⋆. In order to treat the nonindecomposable case, the interval inversion is introduced. The paper concludes with an extension of this result to the digraphs which do not admit as subdigraphs
INDECOMPOSABLE BINARY STRUCTURES
, 2009
"... The notion of interval is wellknown for linear orders. The analogue for (undirected) graphs is called module [25] or homogeneous set [6]. One uses also autonomous set [16, 21, 22] for partially ordered sets. It is still called interval for relations and multirelations [14, 15], and for directed gra ..."
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The notion of interval is wellknown for linear orders. The analogue for (undirected) graphs is called module [25] or homogeneous set [6]. One uses also autonomous set [16, 21, 22] for partially ordered sets. It is still called interval for relations and multirelations [14, 15], and for directed graphs [18, 24]. For 2structures [11, 13], it is called clan. In our framework, it is easier and more efficient to consider labelled 2structures [13], simply called binary structures [19]. Given a binary structure, a quotient is naturally associated with a partition in clans of its vertex set. The notions above were mainly introduced to obtain a simple notion of quotient. A binary structure admitting a nontrivial quotient is decomposable, otherwise it is indecomposable (or prime or primitive).