T. Kloks, D. Kratsch and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Res. Rep. 93-46, Eindhoven University of Technology, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....lattices, by equating the closed sets of the lattice with the set of minimal separators of an underlying graph. The notion of minimal separator, introduced by Dirac in 1961 to characterize chordal graphs (see [10] has been studied extensively during the past decade on non chordal graphs (see [17], 16] 21] 3] 27] and has yielded many new theoretical and algorithmical graph results. We will apply some of these results to analyzing and decomposing a binary relation and the associated concept lattice. Because of space restrictions, we will mostly limit ourselves to presenting our ....

....co atom of L(R) then B is a moplex of GR ; there are no other moplexes in GR . 4 Selecting a sublattice by saturating a minimal separator The process of saturating one minimal separator causes a number of other minimal separators to disappear from the graph; this process was rst introduced by [17], is studied in [22] and its mechanism is described and used in [6] In this Section, we will examine what happens to the lattice when a minimal separator of the underlying graph is saturated. De nition 4.1 ( 17] Let S and T be two minimal separators of graph G; T is said to cross S if there are ....

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T. Kloks, D. Kratsch and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Res. Rep. 93-46, Eindhoven University of Technology, 1993.

....is executed, this causes a number of initial minimal separators to disappear from the graph. Thus, during the process, the set of minimal separators shrinks until it reaches its terminal size of at most n Gamma 1. The minimal separators that disappear are well defined. Kloks, Kratsch and Spinrad [18] introduced the notion of crossing separators , and they showed that a minimal triangulation corresponds to the saturation of a set of non crossing minimal separators. Parra and Scheffler [23] extended this result to characterize minimal triangulations as graphs obtained by saturating a maximal ....

T. Kloks, D. Kratsch, and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Res. Rep. 93-46, Eindhoven University of Technology, 1993.

....concept lattices, by equating the concepts of the lattice with the set of minimal separators of an underlying graph. The notion of minimal separator, introduced by Dirac in 1961 to characterize chordal graphs ( 11] has been extensively studied during the past decade on non chordal graphs (see [22], 21] 28] 3] 38] and has yielded many new theoretical and algorithmical graph results. LIMOS FRE CNRS 2239, Ensemble Scienti que des Czeaux, Universit Blaise Pascal, 63170 Aubire, France. E mail: berry isima.fr, sigayret isima.fr We will apply some of these results to analyzing and ....

.... an atom of L(R) If A O, we prove dually that B A is a co atom of L(R) 11 4 Selecting a sublattice by saturating a minimal separator The process of saturating one minimal separator causes a number of other minimal separators to disappear from the graph; this process was rst introduced by [22] in the context of using the minimal separators of a graph to compute a minimal triangulation. The process is extensively studied in [29] and [28] and its mechanism is described and used in [7] In this Section, we will examine what happens to the lattice when a minimal separator of the ....

[Article contains additional citation context not shown here]

T. Kloks, D. Kratsch and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Res. Rep. 93-46, Eindhoven University of Technology, 1993.

....separators to disappear. Thus during the process, the set of minimal separators shrinks until it reaches the terminal size of less than n. The minimal separators which disappear at some step are well de ned as the separators which cross S, a notion introduced by Kloks, Kratsch and Spinrad ([13] and studied also by Parra ( 17] Let us now examine what happens to moplexes in the course of a triangulating process. Whenever a moplex is made simplicial by the saturation of the minimal separator which is its neighborhood, it will be preserved throughout the entire process. The set of ....

T. Kloks, D. Kratsch, and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Res. Rep. 93-46, Eindhoven University of Technology, 1993.

....separators to disappear. Thus during the process, the set of minimal separators shrinks until it reaches the terminal size of less than n. The minimal separators which disappear at some step are well defined as the separators which cross S, a notion introduced by Kloks, Kratsch and Spinrad ([12] and studied also by Parra ( 16] Let us now examine what happens to moplexes in the course of a triangulating process. Whenever a moplex is made simplicial by the saturation of the minimal separator which is its neighborhood, it will be preserved throughout the entire process. The set of ....

T. Kloks, D. Kratsch, and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Res. Rep. 93-46, Eindhoven University of Technology, 1993.

....the treewidth of a graph [2] For several graph classes, however, there are polynomial time algorithms available for the Treewidth problem. These are cographs, permutation graphs, circle graphs, circular arc graphs, distancehereditary graphs and co comparability graphs of bounded dimension [5, 4, 16, 30, 27, 20]. For co bipartite graphs and hence for co comparability graphs the problem The research of the first author has been supported by the graduate school Algorithmische Diskrete Mathematik by the Deutsche Forschungsgemeinschaft, grant We 1265 2 1. 2 ANDREAS PARRA AND PETRA SCHEFFLER is NP hard ....

....of G corresponds to determining a minimum resp. maximum weighted clique in the comparability graph Sigma(G) This problem can be solved in polynomial time [23] Our approach generalizes the algorithms for Treewidth in permutation graphs and co comparability graphs of bounded dimension [4, 20] and those for Minimum Fill In in bipartite permutation graphs and bounded multi tolerance graphs [29, 26] In the following sections, we demonstrate the method for d trapezoid graphs. 5. d Trapezoid Graphs Let us first recall some order theoretic notions. A partial order is denoted by P = V; ....

T. Kloks, D. Kratsch, and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Technical Report 93/46, Eindhoven University, 1993.

....graphs and caterpillars with hairs of length at most two [25, 2] Treewidth is in P for the restricted versions on cographs, permutation graphs, circle graphs, circular arc graphs, comparability graphs of interval orders, and co comparability graphs of bounded dimension resp. interval dimension [5, 4, 17, 26, 11, 18, 23]. From these results we get polynomial time algorithms that compute the bandwidth of claw free cographs, claw free permutation graphs and claw free co comparability graphs of bounded dimension resp. interval dimension by applying Theorem 12. A very interesting conclusion is gained for the fixed ....

T. Kloks, D. Kratsch, and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Technical Report 93/46, Eindhoven University, 1993.

....Kratsch, T. Kloks, H. Muller distinct connected components. If no proper subset of S is an a; b separator then S is a minimal a; b separator. A minimal separator S is a set of vertices such that S is a minimal a; b separator for some nonadjacent vertices a and b. The following lemma was given in [19, 20]. It provides an easy algorithm to recognize minimal separators. Lemma 3.2 Let S be a separator of the graph G = V; E) Then S is a minimal separator of G if and only if there are at least two different connected components of G[V n S] for which every vertex of S has a neighbor in both of these ....

....There is a linear time recognition algorithm for interval graphs by Booth and Lueker which also computes a consecutive clique arrangement of the input graph if it is an interval graph [5] Using the characterization of Lemma 5. 1, we can easily identify the minimal separators of an interval graph [20]. Lemma 5.2 Let G be an interval graph and let A 1 ; A t be a consecutive clique arrangement of G. Then the set of all minimal separators of G consists of the sets S p = A p A p 1 for p 2 f1; 2; t Gamma 1g. Hence an interval graph G = V; E) has at most n minimal separators. ....

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T. Kloks, D. Kratsch and J. Spinrad, Treewidth and pathwidth of cocomparability graphs of bounded dimension, Computing Science Note 93/46, Eindhoven University of Technology, Eindhoven, The Netherlands, 1993.

....T. Kloks, H. Muller distinct connected components. If no proper subset of S is an a; b separator then S is a minimal a; b separator. A minimal separator S is a set of vertices such that S is a minimal a; b separator for some nonadjacent vertices a and b. The following lemma was given in [19, 20]. It provides an easy algorithm to recognize minimal separators. Lemma 3.2 Let S be a separator of the graph G = V; E) Then S is a minimal separator of G if and only if there are at least two different connected components of G[V n S] for which every vertex of S has a neighbor in both of these ....

....There is a linear time recognition algorithm for interval graphs by Booth and Lueker which also computes a consecutive clique arrangement of the input graph if it is an interval graph [5] Using the characterization of Lemma 5. 1, we can easily identify the minimal separators of an interval graph [20]. Toughness and scattering number of interval graphs 11 Lemma 5.2 Let G be an interval graph and let A 1 ; A t be a consecutive clique arrangement of G. Then the set of all minimal separators of G consists of the sets S p = A p A p 1 for p 2 f1; 2; t Gamma 1g. Hence an ....

[Article contains additional citation context not shown here]

T. Kloks, D. Kratsch and J. Spinrad, Treewidth and pathwidth of cocomparability graphs of bounded dimension, Computing Science Note 93/46, Eindhoven University of Technology, Eindhoven, The Netherlands, 1993.

....number of minimal s; t separators in this graph is 2 (n Gamma2) 2 . Our listing algorithm has been applied as an important subroutine in O(n 5 R n 3 R 3 ) algorithms computing the treewidth and minimum fill in of a given asteroidal triple free graph with n vertices and R minimal separators [16, 17]. Notice that asteroidal triplefree graphs are a relatively large class of graphs containing cocomparability graphs and permutation graphs. Furthermore it has been suggested in [18] to use a so called separator graph for obtaining polynomial time treewidth and minimum fill in algorithms. ....

Kloks, T., D. Kratsch and J. Spinrad, Treewidth and pathwidth of cocomparability graphs of bounded dimension, Computing Science Notes 93/46, Eindhoven University of Technology, The Netherlands, (1993).

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T. Kloks, D. Kratsch and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Res. Rep. 93-46, Eindhoven University of Technology, 1993.

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T. Kloks, D. Kratsch, and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Res. Rep. 93-46, Eindhoven University of Technology, 1993. 20

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T. Kloks, D. Kratsch and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Res. Rep. 93-46, Eindhoven University of Technology (1993).

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T. Kloks, D. Kratsch, and J. Spinrad. Treewidth and pathwidth of cocomparability graphs of bounded dimension. Res. Rep. 93-46, Eindhoven University of Technology, 1993.

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