Results 1  10
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11
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 87 (13 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 29 (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.
Efficient and Practical Modular Decomposition
 EIGHTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 1997
"... We give a simple recursive algorithm for modular decomposition of undirected graphs that runs in O(n+ma(m;n)) time. Previous algorithms with this bound are of theoretical use only. By adding some data structure tricks, we get a much simpler proof of an O(n+m) bound than was previously available. Key ..."
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Cited by 16 (5 self)
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We give a simple recursive algorithm for modular decomposition of undirected graphs that runs in O(n+ma(m;n)) time. Previous algorithms with this bound are of theoretical use only. By adding some data structure tricks, we get a much simpler proof of an O(n+m) bound than was previously available. Key components of the algorithm are variations of a procedure for finding a depthfirst forest on the complement of a directed graph G in O(n+m) time. This is surprising, given that it takes Ω(n²) time to compute the complement explicitly.
Ordered Vertex Partitioning
, 2000
"... A transitive orientation of a graph is an orientation of the edges that produces a transitive digraph. The modulardecomposition of a graph is a canonical representation of all of its modules. Finding a transitive orientation and finding the modular decomposition are in some sense dual problems. In t ..."
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Cited by 6 (3 self)
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A transitive orientation of a graph is an orientation of the edges that produces a transitive digraph. The modulardecomposition of a graph is a canonical representation of all of its modules. Finding a transitive orientation and finding the modular decomposition are in some sense dual problems. In this paper, we describe a simple O(n+mlogn)algorithm that uses this duality to find both a transitive orientation and the modular decomposition. Though the running time is not optimal, this algorithm is much simpler than any previous algorithms that are not Ω(n²). The bestknown time bounds for the problems are O(n + m), but they involve sophisticated techniques.
Task Graph Performance Bounds Through Comparison Methods
, 2001
"... When a parallel computation is represented in a formalism that imposes seriesparallel structure on its task graph, it becomes amenable to automated analysis and scheduling. Unfortunately, its execution time will usually also increase as precedence constraints are added to ensure seriesparallel str ..."
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Cited by 5 (1 self)
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When a parallel computation is represented in a formalism that imposes seriesparallel structure on its task graph, it becomes amenable to automated analysis and scheduling. Unfortunately, its execution time will usually also increase as precedence constraints are added to ensure seriesparallel structure. Bounding the slowdown ratio would allow an informed tradeoff between the benefits of a restrictive formalism and its cost in loss of performance. This dissertation deals with seriesparallelising task graphs by adding precedence constraints to a task graph, to make the resulting task graph seriesparallel. The weak bounded slowdown conjecture for seriesparallelising task graphs is introduced. This states that the slowdown is bounded if information about the workload can be used to guide the selection of which precedence constraints to add. A theory of best seriesparallelisations is developed to investigate this conjecture. Partial evidence is presented that the weak slowdown bound is likely to be 4/3, and this bound is shown to be tight.
NLC2 decomposition in polynomial time
 In Proceedings of GraphTheoretical Concepts in Computer Science, volume 1665 of LNCS
, 1999
"... NLCk is a family of algebras on vertexlabeled graphs introduced by Wanke. An NLCdecomposition of a graph is a derivation of this graph from single vertices using the operations in question. The width of the decomposition is the number of labels used, and the NLCwidth of the graph is the smallest ..."
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Cited by 4 (1 self)
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NLCk is a family of algebras on vertexlabeled graphs introduced by Wanke. An NLCdecomposition of a graph is a derivation of this graph from single vertices using the operations in question. The width of the decomposition is the number of labels used, and the NLCwidth of the graph is the smallest width among its NLCdecompositions. Many difficult graph problems can be solved efficiently with dynamic programming if an NLCdecomposition of low width is given for the input graph. It is unknown though whether arbitrary graphs of NLCwidth at most k can be decomposed with k labels in polynomial time. So far this has been possible only for k = 1, which corresponds to cographs. In this paper, an algorithm is presented that works for k = 2. It runs in O(n4 log n) time and uses O(n2) space. Related concepts: cliquedecomposition, cliquewidth. Keywords: Graph algebra; graph decomposition; NLCwidth; cliquewidth; cograph. 1
Partially Complemented Representations of Digraphs
, 1999
"... this paper, we explore algorithmic uses of this concept on graphs and digraphs. An outward complementation operation is where only the outgoing arcs of a vertex are complemented. That is, the neighbors of the vertex are turned into nonneighbors and the nonneighbors are turned into neighbors. An in ..."
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Cited by 4 (0 self)
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this paper, we explore algorithmic uses of this concept on graphs and digraphs. An outward complementation operation is where only the outgoing arcs of a vertex are complemented. That is, the neighbors of the vertex are turned into nonneighbors and the nonneighbors are turned into neighbors. An inward complementation operations is where only the inward arcs are complemented. A symmetric complementation operation is one where both the inward and the outward arcs are complemented.
Graph Decomposition Using Node Labels
 Doctoral Dissertation, Royal Institute of Technology
, 2001
"... Dynamic programming is a method in which the solution to a computational problem is found by combination of already obtained solutions to subproblems. This method can be applied to problems on graphs (nodes connected by edges). The graph of interest must then be broken down into successively smaller ..."
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Cited by 3 (0 self)
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Dynamic programming is a method in which the solution to a computational problem is found by combination of already obtained solutions to subproblems. This method can be applied to problems on graphs (nodes connected by edges). The graph of interest must then be broken down into successively smaller parts according to some suitable principle. The thesis studies two graph algebras introduced for this purpose (Wanke 1994; Courcelle and Olariu 1994). Nodes have labels, and roughly, the operations are union, edge drawing, and relabeling. Once two nodes have acquired the same label, they can no longer be differentiated at subsequent edge drawings and relabelings.
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...
An Algorithm for the Modular Decomposition of Hypergraphs
"... We propose an O(n ) algorithm to build the modular decomposition tree of hypergraphs of dimension 3 and show how this algorithm can be generalized to compute in O(n ) time the decomposition of hypergraphs of any fixed dimension k. ..."
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Cited by 1 (0 self)
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We propose an O(n ) algorithm to build the modular decomposition tree of hypergraphs of dimension 3 and show how this algorithm can be generalized to compute in O(n ) time the decomposition of hypergraphs of any fixed dimension k.