N. Robertson, P. D. Seymour, Graph minors ii: algorithmic aspects of treewidth, Journal of Algorithms 7 (1986) 309--322.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....no H minor has bounded tree width. 8 Sketch proof. For n 1 let the n grid be a graph with vertex set f(i; j) 1 i; j ng and two vertices (i; j) and (i ; j ) are connected by an edge if and only if ji Gamma i j jj Gamma j j = 1. It can be shown that the n grid has tree width n [16]. The tree width theorem [17] shows that for each n there exists an m(n) 1 such that every graph with tree width m(n) has an n grid minor. Hence a graph has large tree width if and only if it has a large grid minor. Let H be a planar graph. There is a planar graph H with maximum degree 3 ....

N. Robertson and P. D. Seymour. Graph minors II : algorithmic aspects of tree-width. Journal of Algorithms, 7:309--322, 1986. 15

....# acyclicity is the hereditary version of # acyclicity and far less general. A similar result for # acyclic prime CNFs was left open. We give a positive answer and show that for # acyclic prime #, Dualization is solvable with polynomial delay. Formulas of Bounded Treewidth. The treewidth [41] of a graph expresses its degree of cyclicity. Treewidth is an extremely general notion, and bounded treewidth generalizes almost all other notions of near acyclicity. Following [11] we define the treewidth of a hypergraph resp. monotone CNF # as the treewidth of its associated (bipartite) ....

....and 3. for any variable x i V , the nodes w # W X(w) induce a (connected) subtree of T . The width of T is maxw#W X(w) 1, and the treewidth of #, denoted by Tw 1 (#) is the minimum width over all its tree decompositions. Note that the usual definition of treewidth for a graph [41] results in the case where # is a 2 CNF. Similarly to acyclicity, there are several notions of treewidth for hypergraphs resp. monotone CNFs. For example, tree decomposition of type II of CNF # = c#C c is defined as type I tree decomposition of its incident 2 CNF (i.e. graph) G(#) 11, 20] ....

N. Robertson and P. Seymour. Graph minors II: Algorithmic aspects of tree-width. J. Algorithms, 7:309--322, 1986.

....graph without multiple edges or loops. For a set of vertices V V , the subgraph of G, induced by V is denoted by G[V ] The vertex and edge set of a graph G are denoted by V (G) and E(G) respectively. The notions of treewidth and pathwidth were introduced by Robertson and Seymour in [40] and [39] Definition 1 A tree decomposition of G = V; E) is defined to be a pair (fX i : i 2 Ig; T ) where fX i : i 2 Ig is a collection of subsets of V (we call these subsets the nodes of the decomposition) and T = I; F ) is a tree having the index set I as set of vertices, such that the ....

N. Robertson and P.D. Seymour, "Graph Minors II: Algorithmic aspects of tree-width", J. of Algorithms, 7 (1986), 309--322.

....for acyclic prime CNFs was left open. For non prime acyclic CNFs, this is trivially as hard as the general case. In this paper, we give a positive answer and show that for acyclic (prime) DUALIZATION is solvable with polynomial delay. Formulas of Bounded Treewidth. The treewidth [45] of a graph expresses its degree of cyclicity. Treewidth is an extremely general notion, and bounded treewidth generalizes almost all other notions of near acyclicity. Following [13] we define the treewidth of a hypergraph resp. monotone CNF as the treewidth of its associated (bipartite) ....

....3. for any variable x i 2 V , the set of nodes fw 2 W j x i 2 X(w)g induces a (connected) subtree of T . The width of T is maxw2W jX(w)j 1, and the treewidth of , denoted by Tw 1 ( is the minimum width over all its tree decompositions. Note that the usual definition of treewidth for a graph [45] results in the case where is a 2 CNF. Similarly to acyclicity, there are several notions of treewidth for hypergraphs resp. monotone CNFs. For example, tree decomposition of type II of CNF = c is defined as type I tree decomposition of its incident 2 CNF (i.e. graph) G( 13, 24] That ....

N. Robertson and P. Seymour. Graph minors II: Algorithmic aspects of tree-width. Journal of Algorithms, 7:309--322, 1986.

....trees where the leaves have sibling pointers, structured compiler control flow graphs, and with some extension, rectangular grids and other less tree like structures. In fact, while the graphs we generate are more general than trees, the grammar specification defines an upper limit on tree width [90]; thus they are all tree like in a mathematical sense, and in a manner correlated with the grammar definition. In the following section we formalize the notion of partitionability and describe the criteria we will use for measuring quality. Section 3 develops the basis for the grammars in ....

....grid can be generated by this grammar in w h steps, s = w h. By Lemma 52, and the construction in section 7.2.1 the upper bound on tree width follows. # 42 Corollary 55 jibes nicely with existing results; it is known that square grids of # n # n vertices (n # 2) have a tree width of # n [90]. 9 Related Work In the interests of generality, we have investigated graph partitioning under the assumption that any number of partitions may be demanded. However, related problems such as determining the minimum number of edges to be cut to separate a graph into just k partitions for a fixed ....

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N. Robertson and P. D. Seymour. Graph minors II: Algorithmic aspects of treewidth. Journal of Algorithms, 7:309--322, 1986.

....Vertex Multicut. We have a proof that Vertex Multicut is NP hard in bounded degree trees. Unrestricted Vertex Multicut is easier: it is polynomially solvable in trees, but it becomes NP hard in bounded degree series parallel graphs. The tree width notion (first introduced by Robertson and Seymour [22, 21]) seems to often capture a property of the input graph which makes hard problems easy. Various NP hard problems, like Clique or Coloring, have a polynomial time algorithm (linear time in fact) if the input graph has bounded tree width (see for example [2] We will present the formal definition of ....

N. Robertson and P. Seymour, "Graph Minor II. Algorithmic Aspects of Tree-Width," Journal of Algorithms, 7, 309--322, 1986.

....and the connectivity of a system structure is given in [Darwiche 1998b] for the interactive reasoning mode and in [Darwiche 1998a, Darwiche 1999a] for the embeddable mode. In a nutshell, the system connectivity is measured using a graphtheoretic parameter known as the treewidth [Dechter 1992, Robertson and Seymour 1986, Arnborg 1985] The cnets algorithms are known to be exponential only in the treewidth of the system structure and linear in all other aspects of the system. Therefore, if the treewidth is bounded, then compiling a device can be accomplished in linear time and leads to a compilation which can be ....

Robertson, N. and Seymour, P. D. 1986. Graph minors II: Algorithmic aspects of tree-width. Journal of Algorithms, 7:309--322. 19

....Vertex Multicut. We have a proof that Vertex Multicut is NP hard in bounded degree trees. Unrestricted Vertex Multicut is easier: it is polynomially solvable in trees, but it becomes NP hard in bounded degree series parallel graphs. The tree width notion (first introduced by Robertson and Seymour [RS84, RS86]) seems to often capture a property of the input graph which makes hard problems easy. Various NP hard problems, like Clique or Coloring, have a polynomial time algorithm (linear time in fact) if the input graph has bounded tree width (see for example [AL91] We will present the formal ....

N. Robertson and P. Seymour, "Graph Minor II. Algorithmic Aspects of Tree-Width," Journal of Algorithms, 7, 309--322, 1986.

....at most l elements of S, the nodes in which an element of S participates form a subtree of t, and each relation occurrence in S involves elements contained in a single node of t. In the literature, when S is a graph and S is an (l; k) tree, then it is said to have tree width k Gamma 1 (see e.g. [34, 43]) Along the general lines of duality of graph homomorphisms (see Hell, Nesetril and Zhu [23] a constraint satisfaction problem defined by a template T has (l; k) tree duality if: A structure S can be mapped to T if and only if every (l; k) tree that can be mapped to S can be mapped to T . The ....

N. Robertson and P. Seymour, "Graph minors. II. Algorithmic aspects of treewidth, " J. of Algorithms 7 (1985), 309--322.

....properties of graphs can be verified by induction on the tree, often leading to efficient algorithms. 1 Introduction A lot of efforts have been made in the last decade to obtain small specifications of graphs. A well supported idea has been that of representing graphs by expressions or trees [1, 6, 12]. Recently, A. Ehrenfeucht et al. 9] have introduced a representation of finite graphs by finite prefix free languages of strings whose alphabets have themselves a graph structure. The strings of the language represent the vertices and there is an edge between two vertices if and only if the pair ....

N. Robertson and P. Seymour, "Graph Minors. II Algorithmic aspects of treewidth ", Journal of Algorithms, 7 (1986) 309--322.

....paths problem can now be solved in O(n 2 ) time. This is the primary motivation for this paper. Keywords: Tree decompositions, treewidth, disjoint rooted paths. 1. Introduction The notions of tree decompositions and treewidth of a graph were introduced in the 1980 s by Robertson and Seymour [17] and they have since found a large number of applications (see Bodlaender [12] for a tutorial on graph treewidth and its applications) We shall define tree decompositions and treewidth precisely in the next section. For now it will suffice to think of a tree decomposition of a graph G as a ....

....can be solved in polynomial time (for references see [12] This is what spurred an interest in finding tree decompositions of small (constant) width in graphs which have them. The first algorithm for finding an O(k) tree decomposition (for some constant k) was due to Robertson and Seymour [17] and had an O(n f(k) running time on graphs with n vertices (Robertson and Seymour never bothered to compute f(k) This was first improved by Arnold, Corneil and Proskurowski [10] who gave an O(n k 2 ) algorithm, then again by Robertson and Seymour [19] who gave an O(n 2 ) algorithm, and ....

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N. Robertson and P. D. Seymour, "Graph Minors II. Algorithmic aspects of treewidth, " J. Algorithms 7 (1986) 309--322

....to O(ff(n) where ff(n) is the inverse of Ackermann s function [1] and is a very slowly growing function. Another important subclass of sparse graphs is the class of graphs with bounded treewidth. The study of graphs using the treewidth as a parameter was pioneered by Robertson and Seymour [17] and continued by many others (see e.g. 4, 5, 6, 7] Roughly speaking, the treewidth of a graph G is a parameter which measures how close is the structure of G to a tree. A formal definition is given in Section 2. Graphs of treewidth t are also known as partial t trees and have at most tn ....

....where T b is a binary tree and jV (T b )j 2(n Gamma t) 5 Part (b) of the above fact follows by the usual binarization of an arbitrary tree. We will use this in Section 5. Given a tree decomposition of G, we can quickly find separators in G, as the following proposition shows. Proposition 2. 1 [17] Let G be a graph and let (X; T ) be its tree decomposition. Also let e = i; j) 2 E(T ) and let T 1 and T 2 be the two subtrees obtained by removing e from T . Then X i X j separates S m2V (T1 ) Xm Gamma (X i X j ) from S m2V (T2 ) Xm Gamma (X i X j ) 3 Constructing a shortest path ....

N. Robertson and P. Seymour, Graph Minors II: Algorithmic Aspects of Treewidth, J. of Algorithms, 7 (1986), 309-322. 17

....linear time. Keywords: combinatorial problems, algorithms, analysis of algorithms 1 Introduction The topics of the pathwidth and treewidth of graphs have proven to be of fundamental interest for two reasons. First of all, they play an important role in the deep results of Robertson and Seymour [20, 21, 22, 24]. Secondly, and more importantly from a practical point of view, bounded pathwidth and treewidth have proven to be general common denominators for many natural input restrictions of NP complete problems. For many important problems, we now know that fixing a natural parameter k implies that the ....

N. Robertson and P. D. Seymour. Graph minors II: algorithmic aspects of treewidth. J. Algorithms 7 (1986), 309--322.

....5 below) However we give the following direct proof to make explicit the dependence on the diameter, and to show that the result does not introduce any of the scary constants ubiquitous in Robertson Seymour theory. We first define the concept of tree width, introduced by Robertson and Seymour [35] and now standard in graph theory. Definition 1. A tree decomposition of a graph G is a representation of G as a subgraph of a chordal graph G # . The width of the tree decomposition is one less than the size of the largest clique in G # . The tree width of G is the minimum width of any tree ....

N. Robertson and P. D. Seymour. Graph minors II: algorithmic aspects of tree-width. J. Algorithms, 7:309--322, 1986.

....result concerns the case of bounded treewidth networks. Informally, the treewidth t is a parameter that indicates how close the structure of the network is to a tree (see Section 2 for a formal definition) The class of bounded treewidth networks pioneered by the work of Robertson and Seymour [9] and continued by many other researchers (see e.g. 3, 4, 5] includes (among others) outerplanar networks (t = 2) series parallel networks (t = 2) and networks with bounded bandwidth or cutwidth [3, 5] For the case of k terminal bounded treewidth networks, we show that there is a mimicking ....

N. Robertson and P. Seymour, "Graph Minors II: Algorithmic Aspects of Treewidth", J. of Algorithms 7 (1986), pp.309-322.

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N. Robertson, P. D. Seymour, Graph minors ii: algorithmic aspects of treewidth, Journal of Algorithms 7 (1986) 309--322.

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N. Robertson and P. D. Seymour. Graph minors ii: algorithmic aspects of treewidth. Journal of Algorithms, 7:309322, 1986.

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N. Robertson and P.D. Seymour, Graph Minors II: Algorithmic aspects of tree-width, J. Algorithms 7 (1986), 309-322.

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ROBERTSON, N., AND SEYMOUR, P. D. Graph minors ii: Algorithmic aspects of tree-width. Journal of Algorithms, 7 (1986), 309--322.

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N. Robertson, and P.D. Seymour. Graph minors II: algorithmic aspects of tree-width. J. Algorithms, 7:309--322,1986.

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N. Robertson, and P.D. Seymour. Graph minors II: algorithmic aspects of tree-width. J. Algorithms, 7:309--322,1986.

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N. Robertson, and P.D. Seymour. Graph minors II: algorithmic aspects of tree-width. J. Algorithms, 7:309--322,1986.

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N. Robertson and P. Seymour, "Graph Minor II. Algorithmic Aspects of Tree-Width," Journal of Algorithms, 7, 309--322, 1986.

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N. Robertson and P.D. Seymour, Graph minors II: algorithmic aspects of treewidth, J. of Algorithms 7 (1986) 309-322.

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Robertson, N. and Seymour, P. D., "Graph Minors. II. Algorithmic Aspects of Treewidth," J. Algorithms, 7(1986), pp. 309--322.

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