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21
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.
The modular decomposition of countable graphs: Constructions in Monadic SecondOrder Logic
 IN COMPUTER SCIENCE LOGIC 2005, VOLUME 3634 OF OXFORD, LEC. NOTES COMPUT. SCI
, 2005
"... We show that the modular decomposition of a countable graph can be defined from this graph, given with an enumeration of its set of vertices, by formulas of Monadic SecondOrder logic. A second main result is the definition of a representation of modular decompositions by a low degree relational st ..."
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Cited by 5 (3 self)
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We show that the modular decomposition of a countable graph can be defined from this graph, given with an enumeration of its set of vertices, by formulas of Monadic SecondOrder logic. A second main result is the definition of a representation of modular decompositions by a low degree relational structures, also constructible by Monadic SecondOrder formulas.
A Graph Parsing Algorithm and Implementation
, 1993
"... This paper presents an algorithm for decomposing directed acyclic graphs (DAGs) into a heirarchy of subgraphs we call clans. The resulting parse tree is being used to partition and schedule program dependence graphs for efficient concurrent execution on a parallel system. The clans can be identified ..."
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Cited by 2 (2 self)
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This paper presents an algorithm for decomposing directed acyclic graphs (DAGs) into a heirarchy of subgraphs we call clans. The resulting parse tree is being used to partition and schedule program dependence graphs for efficient concurrent execution on a parallel system. The clans can be identified as able or unable to support parallel execution, and their connections with other clans is completely characterized in the parse tree. . 2 Definitions and Concepts
Convex CircuitFree coloration of an oriented graph
 REVISION FOR THE EUROPEAN JOURNAL OF COMBINATORICS
"... We introduce the convex circuitfree coloration and convex circuitfree chromatic number χa ( − → G) of an oriented graph − → G and establish various basic results. We show that the problem of deciding if an oriented graph verifies χa ( − → G) ≤ k is NPcomplete if k ≥ 3 and polynomial if k ≤ 2. We ..."
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Cited by 1 (0 self)
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We introduce the convex circuitfree coloration and convex circuitfree chromatic number χa ( − → G) of an oriented graph − → G and establish various basic results. We show that the problem of deciding if an oriented graph verifies χa ( − → G) ≤ k is NPcomplete if k ≥ 3 and polynomial if k ≤ 2. We exhibit an algorithm which finds the optimal convex circuitfree coloration for tournaments, and characterize the tournaments that are vertexcritical for the convex circuitfree coloration.
Partially critical indecomposable graphs
, 2007
"... Given a graph G = (V, E), with each subset X of V is associated the subgraph G(X) of G induced by X. A subset I of V is an interval of G provided that for any a,b ∈ I and x ∈ V \ I, {a, x} ∈ E if and only if {b, x} ∈ E. For example, ∅, {x}, where x ∈ V, and V are intervals of G called trivial inte ..."
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Given a graph G = (V, E), with each subset X of V is associated the subgraph G(X) of G induced by X. A subset I of V is an interval of G provided that for any a,b ∈ I and x ∈ V \ I, {a, x} ∈ E if and only if {b, x} ∈ E. For example, ∅, {x}, where x ∈ V, and V are intervals of G called trivial intervals. A graph is indecomposable if all its intervals are trivial; otherwise, it is decomposable. Given an indecomposable graph G = (V, E), consider a proper subset X of V such that X  ≥ 4 and G(X) is indecomposable. The graph G is critical according to G(X) if for every x ∈ V \ X, G(V \ {x}) is decomposable. A graph is partially critical if it is critical according to one of its indecomposable subgraphs containing at least 4 vertices. In this paper, we characterize the partially critical graphs.
revision for the European Journal of Combinatorics Convex CircuitFree coloration of an oriented graph
"... We introduce the Convex CircuitFree coloration and Convex CircuitFree chromatic number −→χa( G) of an oriented graph G and establish various basic results. We show that the problem of deciding if an oriented graph verifies χa( G) ≤ k is NPcomplete if k ≥ 3 and polynomial if k ≤ 2. We exhibit an ..."
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We introduce the Convex CircuitFree coloration and Convex CircuitFree chromatic number −→χa( G) of an oriented graph G and establish various basic results. We show that the problem of deciding if an oriented graph verifies χa( G) ≤ k is NPcomplete if k ≥ 3 and polynomial if k ≤ 2. We exhibit an algorithm which finds the optimal convex circuitfree coloration for tournaments, and characterize the tournaments that are vertexcritical for the convex circuitfree coloration.
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.
Indecomposable Graphs and Signed Domination Numbers
, 2005
"... Graph Theory has a long and rich history. It has provided a multitude of avenues of study for more than two centuries and will do so for centuries to come. Graph theory offers numerous interesting theoretical as well as applicationoriented problems. One such problem is the traveling salesman ..."
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Graph Theory has a long and rich history. It has provided a multitude of avenues of study for more than two centuries and will do so for centuries to come. Graph theory offers numerous interesting theoretical as well as applicationoriented problems. One such problem is the traveling salesman