A. Parra and P. Scheer. How to use the minimal separators of a graph for its chordal triangulation. Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP '95), Lecture Notes in Computer Science, 944:123-134, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 two di erent ....

....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 two di erent connected components C 1 and C 2 of G(V S) such that T C 1 6= and T C 2 6= Property 4. 2 [22] Let G be a graph, let S be a minimal separator of G, let GS denote the graph obtained from G by saturating S; then T is a minimal separator of GS i T is a minimal separator of G and T does not cross S in G. We will use this result on our underlying graph GR : saturating a minimal separator S of ....

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A. Parra and P. Scheer. How to use the minimal separators of a graph for its chordal triangulation. Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP '95), Lecture Notes in Computer Science, 944:123134, 1995.

....G = V; E) a vertex set S V is a minimal separator if G(V S) has at least two connected components C 1 and C 2 such that NG (C 1 ) NG (C 2 ) S (C 1 and C 2 are called full components) Characterization 2.2 ( 8] A graph is chordal i all its minimal separators are cliques. Recent research [14, 21, 2] has shown that minimal separators are central to minimal triangulations. The idea behind this is that forcing a graph into respecting Dirac s characterization will result into a minimal triangulation, by repeatedly choosing a not yet processed minimal separator and saturating it. We will need the ....

....will refer to this generalized process as the Saturation Algorithm. Given a set S of minimal separators of G, we will denote G S the graph obtained from G by saturating all the separators belonging to S. The following results from the works of Kloks, Kratsch and Spinrad [14] and Parra and Sche er [21] provide a proof of this algorithm and will be used in Sections 3 and 4. Theorem 2.4 ( 21] A graph H = V; E F ) is a minimal triangulation of G = V; E) i there is a maximal set of pairwise non crossing minimal separators of G such that H = G S . Corollary 2.5 A graph H = V; E F ) is a ....

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A. Parra and P. Scheer. How to use the minimal separators of a graph for its chordal triangulation. Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP '95), Lecture Notes in Computer Science, 944:123-134, 1995.

....techniques force the graph into respecting Fulkerson and Gross characterization, but recent approaches have been made in the direction of forcing the graph into respecting Dirac s characterization. Recent research has shown that minimal triangulation is closely related to minimal separation [2, 19, 23, 30]: the process of repeatedly choosing a minimal separator and adding edges to make it into a clique until all the minimal separators of the resulting graph are cliques, will compute a minimal triangulation. Conversely, any minimal triangulation can be obtained by some instance of this process. A ....

....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 set of pairwise non crossing minimal separators. Definition 3.5 (Kloks, Kratsch, and Spinrad [19] Let S and T be two minimal separators of G. Then S crosses T if there exist two components C 1 ....

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A. Parra and P. Scheffler. How to use the minimal separators of a graph for its chordal triangulation. Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP '95), Lecture Notes in Computer Science, 944:123--134, 1995.

....a one to one correspondence between the concepts of the lattice and the minimal separators of the graph. This is algorithmically interesting because, in the past decade, much research has been done on using minimal separators to eciently solve various graph problems such as chordal embedding ([19], 2] and in particular several papers deal with the ecient enumeration of minimal separators ( 14] 21] 20] 3] 4] pointed out that, using the underlying co bipartite graph and these recent results on the emerging theory of minimal separation, the current best algorithms for generating ....

A. Parra and P. Scheer. How to use the minimal separators of a graph for its chordal triangulation. Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP '95), Lecture Notes in Computer Science, 944:123-134, 1995.

....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 underlying graph is saturated. De nition 4.1 ( 22] Let S and T be two minimal separators of graph G; T is said to cross S if there are two ....

....separators of graph G; T is said to cross S if there are two di erent connected components C 1 and C 2 of G(V S) such that T C 1 6= and T C 2 6= Theorem 4.2 ( 28] A minimal separator of a graph G is a clique separator i it does not cross any other minimal separator of G. Property 4. 3 ([29]) Let G be a graph, let S be a minimal separator of G, let G S denote the graph obtained from G by saturating S; then T is a minimal separator of G S i T is a minimal separator of G and T does not cross S in G. We will use this result on our underlying graph GR : saturating a minimal separator S ....

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A. Parra and P. Scheer. How to use the minimal separators of a graph for its chordal triangulation. Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP '95), Lecture Notes in Computer Science, 944:123 134, 1995. 24

.... it has been shown that the treewidth can be computed in polynomial time, as e.g. cographs [9] circular arc graphs [35] chordal bipartite graphs [23] permutation graphs [10] circle graphs [19] cocomparability graphs of bounded dimension [25] cointerval graphs [16] and d trapezoid graphs [31]. The algorithm for d trapezoid graphs assumes that a d trapezoid intersection model is part of the input. In this paper we present an O(n tw(G) algorithm finding optimal tree and path decompositions for d trapezoid graphs, d a fixed positive integer, where a d trapezoid diagram is part of ....

....model is part of the input. In this paper we present an O(n tw(G) algorithm finding optimal tree and path decompositions for d trapezoid graphs, d a fixed positive integer, where a d trapezoid diagram is part of the input. Note that the best timebound known up to now is O(max(n ) [31]. On the other hand, there is an O(n R n R ) algorithm computing the treewidth and pathwidth of a given asteroidal triple free graph on n vertices with R minimal separators [25] This implies that the treewidth and the pathwidth of a d trapezoid graph can be computed by an O(n ) ....

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A. Parra and P. Scheffler, How to use the minimal separators of a graph for its chordal triangulation, Proceedings of the 22nd International Colloquium on Automata, Languages and Programming, 123\Gamma134, Springer-Verlag, Lecture Notes in Computer Science 944, 1995.

....we obtain an important algorithmic consequence of Theorem 14. We show that for fast functions the treecost of graphs with a polynomial number of minimal separators can be computed eciently. Our approach to this problem follows the ideas of Bouchitt e and Todinca [6] See also Parra and Sche er [16]. This allows one to nd the treecost eciently when the input is restricted to cocomparability graphs, d trapezoid graphs, permutation graphs, circle graphs, weakly triangulated graphs and many others graph classes. See [7] for an encyclopedic survey on graph classes. A subset S of vertices of a ....

....a set of minimal separators G we denote by G the graph obtained from G by turning all separators from into cliques. There is a deep relation between the minimal separators of a graph and its minimal triangulations. We need the following generalization of Dirac s theorem by Parra and Sche er [16]. Two separators S and T cross if there are distinct components C and D of G n T such that S intersects both of them. If S and T do not cross, they are called parallel. Theorem 18 ( 16] i) Let G be a maximal set of pairwise parallel separators of G. Then H = G is a minimal triangulation of ....

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A. Parra and P. Scheer. How to use the minimal separators of a graph for its chordal triangulation. In Automata, languages and programming (Szeged, 1995.

....the major structural results of the paper on which our ecient algorithms are based. In Theorem 17 we establish a representation theorem of minimal triangulations of a d trapezoid graph G in terms of scanlines of a d trapezoid diagram D(G) Hence in contrast to previous work in this area as e.g. [9, 20, 28, 32], our algorithms are based on a general representation theorem that enables the design of an algorithm for Treewith, Minimum Fill in and possibly related problems (concerning the optimization of a graph parameter over all minimal triangulations of the graph) A similar representation theorem is ....

.... The algorithm to compute the treewidth and the pathwidth has running time O(n tw(G) The algorithm to compute the minimum ll in and the minimum interval completion has running time O(n ) Up to now the best known algorithms for all four problems had running time O(max(n 2:376d ; n ) [28]. Our algorithms are simple and ecient for trapezoid graphs (d = 2) In that case they do not even require a trapezoid diagram as part of the input. We obtain O(n ) algorithms to compute all the four graph parameters on trapezoid 6 ) of the algorithm in [28] Furthermore we obtain an O(n ....

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A. Parra and P. Scheer, How to use the minimal separators of a graph for its chordal triangulation, Proceedings of the 22nd International Colloquium on Automata, Languages and Programming, 123 134, Springer-Verlag, Lecture Notes in Computer Science 944, 1995.

....compute minimal triangulations. Different algorithms and different proofs are given for each of these classes of graphs. This paper gives a unified version of the cited algorithms. Characterizations of the minimal triangulations of a graph by the minimal separators have already been given in [2] [11]. Their approach is a global vision of all the minimal separators of a graph and therefore they do not yield an algorithmic construction of minimal triangulations with small cliquesize. A more local view of the minimal triangulations has been tried in [6] but it was unfortunately not correct [7] ....

....for particular classes of graphs, including those previously mentioned. In section 2 we present the relation between the minimal triangulations of a graph and its separators. This leads to a global characterization of the minimal triangulations by maximal sets of pairwise parallel separators [11]. In section 3 we give a local characterization of minimal triangulations by means of maximal sets of neighbor separators . We show the relation between the maximal sets of neighbor separators and the maximal cliques of any minimal triangulation of a graph. In section 4 we prove that if all the ....

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A. Parra and P. Scheffler. How to use the minimal separators of a graph for its chordal triangulation. In Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP'95), volume 994 of Lecture Notes in Computer Science, pages 123--134. Springer-Verlag, 1995.

....the major structural results of the paper on which our ecient algorithms are based. In Theorem 17 we establish a representation theorem of minimal triangulations of a d trapezoid graph G in terms of scanlines of a d trapezoid diagram #(G) Hence in contrast to previous work in this area as e.g. [9,20,28, 32], our algorithms are based on a general representation theorem that enables the design of an algorithm for ########, ####### ####### and possibly related problems (concerning the optimization of a graph parameter over all minimal triangulations of the graph) A similar representation theorem is ....

.... to compute the treewidth and the pathwidth has running time O(n tw(G) d## ) The algorithm to compute the minimum ll in and the minimum interval completion has running time O(n d ) Up to now the best known algorithms for all four problems had running time O(max(n #:###d ;n #d## ) [28]. Our algorithms are simple and ecient for trapezoid graphs (d = 2) In that case they do not even require a trapezoid diagram as part of the input. We obtain O(n # ) algorithms to compute all the four graph parameters on trapezoid graphs (compared to running time O(n # ) of the algorithm in ....

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A. Parra and P.Scheer, How to use the minimal separators of a graph for its chordal triangulation, ########### ## ### 22## ############# ########## ## ### ####### ######### ### ###########, 123 # 134, Springer-Verlag, Lecture Notes in Computer Science 944, 1995.

....following result shown in [1] A graph G has tree width at most k if and only if there exists a minimal separator S with at most k vertices such that every connected component of G S augmented by the completely connected separator vertices has tree width at most k. For further relationships see [19]. In order to motivate our paper consider a class C of graphs G = V; E) having at most p(jV j) different minimal separators, where p(n) is a fixed polynomial. For many examples of such classes as permutation graphs, circle graphs, circular arc graphs, and chordal bipartite graphs, the tree width ....

A. Parra and P. Scheffler. How to use the minimal separators of a graph for its chordal triangulation. In Proceedings of the International Colloquium on Automata, Languages and Programming, volume 944 of Lect. Notes in Comput. Sci., pages 123--134. SpringerVerlag, New York/Berlin, 1995.

....and minimum fill in of a given asteroidal triple free graph with n vertices and R minimal separators [16, 17] Notice that asteroidal triple free 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 fillin algorithms. Thereby the vertex set of the separator graph is the set of all minimal separators of the given graph. Typically, applications require our listing algorithm. For listing other types of ....

A. PARRA AND P. SCHEFFLER, How to use the minimal separators of a graph for its chordal triangulation, in Proceedings of the 22rd International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 944, Springer-Verlag, Berlin, 1995, pp. 123--134.

....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. Thereby the vertex set of the separator graph is the set of all minimal separators of the given graph. Typically applications require our listing algorithm. For listing other types of ....

Parra, A. and P. Scheffler, How to use the minimal separators of a graph for its chordal triangulation, Proceedings of the 22rd International Colloquium on Automata, Languages and Programming, Springer-Verlag, Lecture Notes in Computer Science 944, (1995), pp. 123--134.

.... it has been shown that the treewidth can be computed in polynomial time, as e.g. cographs [9] circular arc graphs [35] chordal bipartite graphs [23] permutation graphs [10] circle graphs [19] cocomparability graphs of bounded dimension [25] cointerval graphs [16] and d trapezoid graphs [31]. The algorithm for d trapezoid graphs assumes that a d trapezoid intersection model is part of the input. In this paper we present an O(n tw(G) d Gamma1 ) algorithm finding optimal tree and path decompositions for d trapezoid graphs, d a fixed positive integer, where a d trapezoid diagram ....

....input. In this paper we present an O(n tw(G) d Gamma1 ) algorithm finding optimal tree and path decompositions for d trapezoid graphs, d a fixed positive integer, where a d trapezoid diagram is part of the input. Note that the best timebound known up to now is O(max(n 2:376 d ; n 2d 2 ) [31]. On the other hand, there is an O(n 5 R n 3 R 3 ) algorithm computing the treewidth and pathwidth of a given asteroidal triple free graph on n vertices with R minimal separators [25] This implies that the treewidth and the pathwidth of a d trapezoid graph can be computed by an O(n 3d 3 ....

[Article contains additional citation context not shown here]

A. Parra and P. Scheffler, How to use the minimal separators of a graph for its chordal triangulation, Proceedings of the 22nd International Colloquium on Automata, Languages and Programming, 123\Gamma134, Springer-Verlag, Lecture Notes in Computer Science 944, 1995.

....this case S is a nested separator in G. We say that a separator S of G crosses another separator S 0 if there are vertices u and v in S 0 which are separated by S in G. It is easy to see, that if S crosses S 0 , then S 0 separates some vertices of S, and therefore S 0 crosses S (e.g. [PS95, Theorem 3]) A graph is called triangulated or chordal if it does not contain a chordless cycle of length greater than 3. An induced subgraph of a triangulated graph is triangulated. A simplicial vertex of a graph G = fv 1 ; v n g; E) is a vertex, such that its neighborhood induces a clique in G. ....

....of G, then the following assertions hold. 1. No pair of minimal separators of T cross each other. 2. Every fill in edge in T connects nonadjacent vertices of a minimal separator of G. 3. Every minimal a; b separator of T is also a minimal a; b separator of G. Proof: See, for example, [PS95, Lemma 5 and Theorem 10]. 2 The width of a triangulated graph T is maxK2KT (jKj Gamma 1) where K T is the set of maximal cliques of T . The treewidth of a graph G is the minimal width over all triangulations of G. A minimal triangulation T of G is called k minimal if the width of T is at most k. A minimal ....

[Article contains additional citation context not shown here]

Parra A. and Scheffler P. How to use the minimal separators of a graph for its chordal triangulations. in Proceedings of ICALP95; Lecture Notes in Computer Science, Springer Verlag, #944, pp. 123-134, 1995.

....the major structural results of the paper on which our e cient algorithms are based. In Theorem 17 we establish a representation theorem of minimal triangulations of a d trapezoid graph G in terms of scanlines of a d trapezoid diagram D(G) Hence in contrast to previous work in this area as e.g. [9, 20, 28, 32], our algorithms are based on a general representation theorem that enables the design of an algorithm for Treewith, Minimum Fill in and possibly related problems (concerning the optimization of a graph parameter over all minimal triangulations of the graph) A similar representation theorem is ....

.... to compute the treewidth and the pathwidth has running time O(n tw(G) d 1 ) The algorithm to compute the minimum ll in and the minimum interval completion has running time O(n d ) Up to now the best known algorithms for all four problems had running time O(max(n 2:376d ; n 2d 2 ) [28]. Our algorithms are simple and e cient for trapezoid graphs (d = 2) In that case they do not even require a trapezoid diagram as part of the input. We obtain O(n 2 ) algorithms to compute all the four graph parameters on trapezoid graphs (compared to running time O(n 6 ) of the algorithm in ....

[Article contains additional citation context not shown here]

A. Parra and P. Scheer, How to use the minimal separators of a graph for its chordal triangulation, Proceedings of the 22nd International Colloquium on Automata, Languages and Programming, 123 134, Springer-Verlag, Lecture Notes in Computer Science 944, 1995.

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A. Parra and P. Scheer. How to use the minimal separators of a graph for its chordal triangulation. Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP '95), Lecture Notes in Computer Science, 944:123-134, 1995.

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A. Parra and P. Scheer. How to use the minimal separators of a graph for its chordal triangulation. Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP '95), Lecture Notes in Computer Science, 944:123-134, 1995.

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A. Parra and P. Scheer. How to use the minimal separators of a graph for its chordal triangulation. Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP '95), Lecture Notes in Computer Science 944 (1995) 123134.

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A. Parra and P. Scheer. How to use the minimal separators of a graph for its chordal triangulation. In Z. Fulop and F. Gecseg, editors, Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP '95), Automata, Languages and Programming, LNCS 944, pages 123-134. Springer Verlag, 1995.

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A. Parra and P. Scheer, \How to use the minimal separators of a graph for its chordal triangulation", Proceedings of the 22nd International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 944, 1995, pp. 123-134.

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