R.H. Mohring, Graph problems related to gate matrix layout and PLA folding. Computational graph theory, Comput. Suppl. 7, 17--51 (1990).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....problems to A, B and C, respectively, with the only change that the supergraph G is required to be interval instead of unit interval. Problem A equivalent to asking if the pathwidth of G is at most k Gamma 1, and it arises in various guises (and under different names) in numerous areas (cf. [33]) It is NP hard [28, 2, 35, 24] but linearly solvable for fixed k [3, 30] Problem C (and hence also B ) was shown to be NP hard when k = jV j independently in [21] and [13] Fellows et al. 13] have also shown that the Problem C hard for W [1] and that it is not finite state for ....

....problem is to decide for a given graph G and a given integer k if pw(G) k. Equivalent problems arise in various areas, including VLSI layout, processor management, node searching and vertex separation (see the excellent survey of Mohring on pathwidth and the relations among these problems [33]) PATHWIDTH is NP complete on arbitrary graphs [28, 2] and even for chordal graphs [24] and (using the equivalence to node search [29] for planar graphs with vertex degrees at most three [35] On the other hand, it is polynomial when k is fixed. The results of Robertson and Seymour [39] imply ....

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R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In G. Tinhofer et al., editors, Computational Graph Theory, Computing Supplement 7, pages 17--51. Springer, Vienna, 1990.

....with linear width. So far, such a linear time algorithm has been constructed (see [3, 4] only for the parameters of treewidth and pathwidth (actually, the result in [3, 4] can be directly transfered to the node search number which is known to be equal to the node search number minus one see [12, 13, 17]) To be precise, 3, 4] state that for fixed k, one can determine in linear time whether a given graph has pathwidth at most k, and if so, find a path decomposition of minimum width. This algorithm first finds a tree decomposition of width at most k, if existing (and if not existing, then the ....

.... ns(G) ms(G ) We mention that the mixed search number is equivalent with the parameter of properpathwidth defined by Takahashi, Ueno, and Kajitani in [24] It is also known that the node search number is equal to the pathwidth, the interval thickness, and the vertex separation number (see [13, 14, 17, 12, 8]) The following is a generalisation of Theorem 2. c) and has been proved in [25] Theorem 3 If G is the graph occurring from G after subdividing each of its pendant edges, then ms(G) linear width(G ) 4.2 Final Comments The result of Theorem 1 has several consequences. First, as one ....

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In E. Mayr, H. Noltemeier, and M. Sys/lo, editors, Computational Graph Theory, Comuting Suppl. 7, pages 17--51. Springer Verlag, 1990.

....algorithm checking membership in G[f; k] where f is ms, es, ns, or linear width. Such a linear time algorithm has been constructed for the node search number [5] actually, the result in [5] concerns the parameter of pathwidth which is known to be equal to the node search number minus one see [21, 19, 25]) Recently, a linear algorithm, checking if a graph belongs to G[linear width; k] was found (see [7] Moreover, the algorithm in [7] is constructive: for any fixed k, one can construct, if exists, an optimal edge arrangement. On the other hand, the algorithm in [7] appears to be difficult to be ....

.... = ms(G ) We mention that the mixed search number is equivalent with the parameter of properpathwidth defined by Takahashi, Ueno, and Kajitani in [33, 35] It is also known that the node search number is equal to the pathwidth, the interval thickness, and the vertex separation number (see [21, 22, 25, 19, 12]) 5.2 The relation between linear width and mixed search A pendant vertex is called fully pendant when it is adjacent with an almost pendant vertex. Any edge containing a fully pendant vertex is called fully pendant. Clearly, a pendant edge is fully pendant iff it is not small. Let G be a ....

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In E. Mayr, H. Noltemeier, and M. Sys/lo, editors, Computational Graph Theory, Comuting Suppl. 7, pages 17--51. Springer Verlag, 1990.

....of G of minimum width. 1 Introduction The notions of pathwidth and treewidth play an important role in many different fields of computer science, often with different terminologies, e.g. ffl Choleski factorization and Gauss elimination. See e.g. 20] ffl VLSI layout theory. See e.g. [34]. ffl theory of expert systems. See e.g. 32] ffl algorithmic graph theory. ffl theory of graph grammars. See e.g. 23] A preliminary version of this paper appeared as Better algorithms for the pathwidth and treewidth of graphs, in the proceedings of ICALP 91. Department of ....

.... This solves an open problem from [16] So far, the only classes of graphs of bounded treewidth for which the complexity of the pathwidth problem was determined (besides classes of graphs with bounded pathwidth) were the trees and the forests: for these the pathwidth can be computed in linear time [21, 34, 43]. 2 Definitions and Preliminary Results The notions of treewidth and pathwidth were introduced by Robertson and Seymour [38, 40] A tree decomposition of a graph G = V; E) is a pair (fX i j i 2 Ig; T = I; F ) with fX i j i 2 Ig a collection of subsets of V , and T = I; F ) a tree, such ....

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R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In E. Mayr, H. Noltemeier, and M. Sys/lo, editors, Computational Graph Theory, Comuting Suppl. 7, pages 17--51. Springer Verlag, 1990.

....graph (G) can be equal to (G) 1 Introduction Let C be a cycle in a graph G. A chord of C is an edge of G joining two vertices of C which are not consecutive. A graph G is called a chordal graph i every cycle in G of length 4 or more has a chord. Chordal graphs arise in many applications (see [7, 9, 13]) which includes the study of evolutionary trees [1] facility location [4] scheduling problems [10] and the solution of sparse systems of linear equations [11, 12] Chordal graphs constitute one of the most important subclasses of perfect graphs [7] The various names given to chordal graphs, ....

R.H.Mohring, Graph Problems Related To Gate Matrix Layout And PLA Folding. Computational Graph Theory, Pages 17-52, Springer, Wein, New York,1990.

....problem [5] and also from the elusiveness of results relating several graph parameters based on linear layouts of graphs. Pathwidth is a graph parameter closely associated with interval graphs, of importance to both algorithmic and structural graph theory, and with applications to VLSI layout [6, 11]. Bandwidth is another widely studied graph parameter, with applications to sparse matrices [1] and notorious for the difficulty of its computation even for trees [3] Recently, in a study of problems motivated by molecular biology, Kaplan and Shamir [5] showed a somewhat surprising connection ....

....Definition 4.3 Given a path decomposition X 1 ; Xm of a graph G, the interval model corresponding to it is given by l(v) minfi : v 2 X i g and r(v) maxfi : v 2 X i g. We say the path decomposition specifies an embedding of G into the interval graph with this interval model. Theorem 4. 4 [6] G is the subgraph of an interval graph with maximum clique size at most k 1 if and only if G has pathwidth at most k. In a recent paper, Kaplan and Shamir [5] show a connection between bandwidth and interval graphs. Theorem 4.5 [5] G is the subgraph of a 0 proper interval graph with maximum ....

R. Mohring, Graph problems related to gate matrix layout and PLA folding, in Computational Graph Theory, Computing Suppl. 7 pp.17-51, SpringerVerlag, 1990.

....graph searching games (see, for instance, 6] to the minimum stack traversal problem [1] and to the pathwidth parameter, discussed in the next section. Indeed, the algorithm we now present to compute this parameter uses a traversal strategy resembling closely that of [1] as well as those used by [12, 9] to compute the pathwidth of a tree. The algorithm will consist of a single bottom up phase, which will start at the degree one vertices of the tree (leaves) and end in a root r with minimum value of tabreq(T r ) over all vertices of T ) For each vertex v, we keep track of larg(v) and ....

....we remark that the definition of treewidth is similar, except that the bags are nodes of a tree, as opposed to being nodes of a path (sequence) and the bags containing a given vertex induce a connected subtree. There is a lineartime algorithm computing pathwidth and path decomposition of a tree, [9], which uses a traversal similar to our minimum table requirement algorithm, based on the following Theorem: Theorem 3.2 [12] For a tree T , pw(T ) k if and only if there exists a vertex v in T such that T n fvg has at least three components with pathwidth at least k. We will use the following ....

R. Mohring, Graph problems related to gate matrix layout and PLA folding, in Computational Graph Theory, Computing Suppl. 7, Springer-Verlag, 17-51, 1990.

....[1] Pathwidth has important applications in the theory of VLSI layout, and is equivalent to several other graph parameters, including minimum chromatic number of a containing interval graph and node search number. The pathwidth problem is also equivalent to the gate matrix layout problem. See [24] for an overview. In Section 3 we show that optimal treewidth, pathwidth, elimination 2 Table 1: Parameters of graph G = V; E) or symmetric sparse matrix A. k tree number smallest k such that G is a subgraph of a k tree treewidth minimum width of a tree decomposition of G pathwidth minimum ....

....i g; T ) such that T is a path. The pathwidth of such a path decomposition is max jX i j Gamma 1. The pathwidth of a graph is the minimum pathwidth over all possible path decompositions of that graph. The notion of pathwidth has several important applications, for example in VLSI layout theory [24]. The next few definitions measure the difficulty of splitting a graph approximately in half by deleting edges or vertices. Let ff be a constant between 0 and 1. An ff vertex separator of a graph G = V; E) is a set S V of vertices such that every connected component of the graph G[V Gamma S] ....

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In G. Tinhofer et al., editor, Computing Supplementum, volume 7, pages 17--51. Springer-Verlag, 1990.

....all proper path decompositions. The analogous notion when condition (4) is removed defines the pathwidth of a graph, originally introduced by Robertson and Seymour [25] It is well known that the pathwidth of a graph is one less than the least clique size of any interval supergraph of it (cf. [19]) The following analogous result holds for proper pathwidth: Observation 3.1 ppw(G) minf (G 0 )jG G 0 and G 0 is proper intervalg Gamma 1. Thus Problem A is equivalent to deciding whether the proper pathwidth of G is not greater than k Gamma 1. A layout of a graph G = V; E) with ....

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In G. Tinhofer et al., editors, Computational Graph Theory, Computing Supplement 7, pages 17--51. Springer, Vienna, 1990.

....with theoretical and practical value. Its closer relation to the black white pebble game of [6] constitutes the theoretical value in complexity theory. Applications in certain layout problems as Weinberger arrays, gate matrix layout, and PLA folding constitute the practical value in VLSI, see [16]. The vertex separation of a graph can be defined using linear layouts, see [7] A linear layout, or simply a layout, L of a graph G is a permutation of the vertices of G. Intuitively L describes how the vertices are to be laid out along a horizontal line. The i th vertex cut of L defines how many ....

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In E. Mayr, H. Noltmeier, and M. Syslo, editors, Computational Graph Theory, pages 17--51. Springer Verlag, 1990.

....to A, B and C, respectively, with the only change that the supergraph G 0 is required to be interval instead of unit interval. Problem A 0 is equivalent to asking if the pathwidth of G is at most k Gamma 1, and it arises in various guises (and under different names) in numerous areas (cf. [33]) It is NP hard [28, 2, 35, 24] but linearly solvable for fixed k [3, 30] Problem C 0 (and hence also B 0 ) was shown to be NP hard when k = jV j independently in [21] and [13] Fellows et al. 13] have also shown that the Problem C 0 is hard for W [1] and that it is not finite state for ....

....problem is to decide for a given graph G and a given integer k if pw(G) k. Equivalent problems arise in various areas, including VLSI layout, processor management, node searching and vertex separation (see the excellent survey of Mohring on pathwidth and the relations among these problems [33]) PATHWIDTH is NP complete on arbitrary graphs [28, 2] and even for chordal graphs [24] and (using the equivalence to node search [29] for planar graphs with vertex degrees at most three [35] On the other hand, it is polynomial when k is fixed. The results of Robertson and Seymour [39] imply ....

[Article contains additional citation context not shown here]

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In G. Tinhofer et al., editors, Computational Graph Theory, Computing Supplement 7, pages 17--51. Springer, Vienna, 1990.

....checking membership in G[f; k] where f is ms, es, ns, or linear width. Such a linear time algorithm has been constructed for the node search number [7,8] actually, the result in [7,8] concerns the parameter of pathwidth which is known to be equal to the node search number minus one see [21,24,28]) Recently, a linear time algorithm, checking if a graph belongs to G[linear width; k] was found (see [5] Moreover, the algorithm in [5] is constructive: for any fixed k, one can construct an optimal edge arrangement, if exists. On the other hand, the algorithm in [5] appears to be difficult ....

.... n ) 29 We mention that the mixed search number is equivalent with the parameter of proper pathwidth defined by Takahashi, Ueno, and Kajitani in [37,39] It is also known that the node search number is equivalent to the pathwidth, the interval thickness, and the vertex separation number (see [14,21,24,25,28]) It is not hard to prove that the node search and the linear width can differ by at most one (it appears as exercise in [40] It is also easy to see that the same relation connects mixed search number and linear width (see Theorem 25.iv) For an example of the values linear width, ms, ns, and ....

R.H. Mohring, Graph problems related to gate matrix layout and PLA folding, in: E. Mayr, H. Noltemeier, M. Sys/lo (Eds.), Computational Graph Theory, Computing Suppl. 7, Springer Verlag, 1990, pp. 17--51.

....Pathwidth of a graph is de ned in the following section. The cutwidth of a graph is the maximum number of edges that would be cut if the vertices of the graph are mapped to a path. The problems of minimizing pathwidth or cutwidth of a graph are interesting in some VLSI gate layout problems, see [43, 52]. The problem also appears in minimizing vertex or edge rankings of graphs which are used in manufacturing systems [45] see [6] for more details) In [30, 1, 5] an O(log 2 n) approximation was presented. Planar Completion: The Minimum Drawing Size is de ned as : given a graph G, to provide a ....

R. H. Mohring, \Graph problems related to gate matrix layout and and PLA folding" In Computing Supplementum, G. Tinhofer editor, v7, pages 17-51, Springer Verlag (1990).

....problem [5] and also from the elusiveness of results relating several graph parameters based on linear layouts of graphs. Pathwidth is a graph parameter closely associated with interval graphs, of importance to both algorithmic and structural graph theory, and with applications to VLSI layout [6, 11]. Bandwidth is another widely studied graph parameter, with applications to sparse matrices [1] and notorious for the difficulty of its computation even for trees [3] Recently, in a study of problems motivated by molecular biology, Kaplan and Shamir [5] showed a somewhat surprising connection ....

....of a graph , the interval model corresponding to it is given by E r p Lm: i axz s and gE zmO i xz s . We say the path decomposition specifies an embedding of into the interval graph with this interval model. 170 Andrzej Proskurowski and Jan Arne Telle Theorem 4. 4 [6] is the subgraph of an interval graph with maximum clique size at most if and only if has pathwidth at most . In a recent paper, Kaplan and Shamir [5] show a connection between bandwidth and interval graphs. Theorem 4.5 [5] is the subgraph of a 0 proper interval graph with ....

R. Mohring, Graph problems related to gate matrix layout and PLA folding, in Computational Graph Theory, Computing Suppl. 7 pp.17-51, Springer-Verlag, 1990.

....problem [5] and also from the elusiveness of results relating several graph parameters based on linear layouts of graphs. Pathwidth is a graph parameter closely associated with interval graphs, of importance to both algorithmic and structural graph theory, and with applications to VLSI layout [6, 11]. Bandwidth is another widely studied graph parameter, with applications to sparse matrices [1] and notorious for the difficulty of its computation even for trees [3] Recently, in a study of problems motivated by molecular biology, Kaplan and Shamir [5] showed a somewhat surprising connection ....

....X 1 ; Xm of a graph G, the interval model corresponding to it is given by l(v) minfi : v 2 X i g and r(v) maxfi : v 2 X i g. We say the path decomposition specifies an embedding of G into the interval graph with this interval model. 170 Andrzej Proskurowski and Jan Arne Telle Theorem 4. 4 [6] G is the subgraph of an interval graph with maximum clique size at most k 1 if and only if G has pathwidth at most k. In a recent paper, Kaplan and Shamir [5] show a connection between bandwidth and interval graphs. Theorem 4.5 [5] G is the subgraph of a 0 proper interval graph with maximum ....

R. Mohring, Graph problems related to gate matrix layout and PLA folding, in Computational Graph Theory, Computing Suppl. 7 pp.17-51, Springer-Verlag, 1990.

....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 yes instances have bounded treewidth or pathwidth (for examples see [5, 15, 18]) We also know that many problems can be solved in linear time when the input includes a bounded width path decomposition (or tree decomposition) of the graph (see [1, 2, 4, 12, 25] and [6] for many further references) After several rounds of improvement [23, 17, 19] the best known algorithm ....

....X j . The pathwidth of a path decomposition X 1 ; X 2 ; X r is max 1ir jX i j Gamma1. The pathwidth of a graph G is the minimum pathwidth over all path decompositions of G. Determining pathwidth is equivalent to several VLSI layout problems such as gate matrix layout and vertex separation [18, 14]. It is easy to see that the family of graphs of pathwidth at most t is a lower ideal in the topological (and minor) order. It is also known that those graphs with order n have at most nt Gamma (t 2 t) 2 edges. Let B h denote the complete binary tree of height h and order 2 h Gamma 1. ....

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In G. Tinhofer, E. Mayr, H. Noltemeier and M. Syslo, editors, Computational Graph Theory, Computing Supplementum 7, pages 17--51. Springer Verlag, 1990. 10

.... search appeared as the first variant of edge search and was introduced by Kirousis and Papadimitriou in [14] Finally, mixed search was introduced in [2] and [23] It is worth mentioning that ns(G) Gamma 1 and ins(G) Gamma 1 are equal to the pathwidth and the treewidth of G respectively (see [6,7,12 14,17]) For surveys concerning graph searching and related parameters see [1,3,9] The recontamination question for a search game asks whether it is equivalent to its monotone version, i.e. whether excluding all the non monotone searches, reduces the searchers ability. If the answer is no, we say ....

.... a loopless graph without multiple edges has agile mixed search number at most k iff it has proper pathwidth k or, equivalently, if it is a subgraph of a k path (a k path can be viewed as a k tree that either has k 1 vertices or it has only two simplicial vertices) Moreover, it is known (see [12,17,13]) that the inert node search number of a loopless graph without multiple edges is the least k for which a graph is a subgraph of a k caterpillar (a k caterpillar is a k tree that is also an interval graph for definitions and results on k caterpillars see e.g. 19] Finally, it is possible to ....

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In E. Mayr, H. Noltemeier, and M. Sys/lo, editors, Computational Graph Theory, Computing Suppl. 7, pages 17--51. Springer Verlag, 1990.

.... an interval, which may be interpreted as a time interval, for example, in scheduling [MR89] or as a subsequence in a linear sequence of items, for example, a subsequence of the sequence of bases of a DNA in molecular biology [GKS94] or a sequence of electronic units in linear VLSI layout styles [Moh90, Mul93a]. Since most of these optimization problems are NP hard one would like to have good enumeration approaches to solve them exactly as well as a tool for gaining more structural insights. Both can be achieved by a polyhedral approach that is based on a good (partial) polyhedral description of the ....

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In G. Tinhofer, E. Mayr, H. Noltemeier, and M. Syslo, editors, Computational Graph Theory, pages 17 -- 52. Springer, 1990.

....of P whose performance ratios depend on the dimension and interval dimension of P , respectively. 1 Introduction The pathwidth and treewidth of a graph are two notions with a large number of different applications in many areas like algorithmic graph theory, VLSI layout, and others (see e.g. [1, 3, 16, 17]. One of the main aspects of this broad applicability is the fact that many NP complete problems become polynomially solvable when restricted to classes of graphs with bounded tree or pathwidth [1] Unfortunately, determining the pathwidth or treewidth of a given graph is NP complete [2] This ....

....about the complexity of determining pathwidth and or treewidth for interesting special classes of graphs. It is shown in [8] that determination of the pathwidth stays NP hard when restricted to chordal graphs. Polynomial algorithms for both treewidth and pathwidth exist for the class of trees [16, 18] and cographs [4] Moreover, treewidth equals pathwidth for cographs [4] This equality of pathwidth and treewidth has led to the natural question for which classes of graphs these parameters are identical or at least close. Trivial other such classes are complete graphs and grid graphs. For ....

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In G. Tinhofer, E. Mayr, H. Noltemeier, and M. Sys/lo, editors, Computational Graph Theory, Computing Suppl. 7, pages 17--51. Springer Verlag Wien, 1990.

....function for a given graph has given rise to much interest in recent years, see [7, 21] for a survey. Usually, two versions are considered. First, in the Minimum Fill In problem, we look for a triangulation where the number of additional edges is minimum. It has applications in VLSI design [24] and in sparse matrix factorization [31] Second, the Treewidth problem corresponds to finding a triangulation with the smallest possible maximum clique size. The treewidth of a graph is a parameter that has been especially widely studied since it was defined by Robertson and Seymour [28] The ....

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In G. Tinhofer, E. Mayr, H. Noltemeier, and M. Sys/lo, editors, Computational Graph Theory, pages 17--52. Springer, Wien, New York, 1990.

....average complexity is needed, it can easily be achieved in a last fine tuning step of the implementation. In the last years a similar approach in structural graph theory has proven to be very fruitful in showing existence of efficient worst case algorithms for a large class of problems, cf. e.g. [RS85b, FL85, FL88, Fel89, Moh90, RS85a, FL92]. The Graph Minors Theory developed by Robertson and Seymour gives such algorithms for graph properties that are hereditary with respect to the graph minor relation. It is based on the fact that the properties under consideration are characterized by a finite number of minimal obstructions. The ....

Rolf H. Mohring, Graph problems related to gate matrix layout and PLA folding, Computational Graph Theory (Wien) (G. Tinhofer et al., eds.), Springer-Verlag, Wien, 1990, pp. 17--52.

....transformed by the principles of [86] to a polynomial algorithm for node searching on trees. This requires O(n) time for determining ns(G) and O(nlogn) time for finding the associated search. Different and faster algorithms with much simpler correctness proofs have independently been obtained in [129], 163] see also [164] The algorithms peel the tree, i.e. they start with the leaves and works its way towards the center of the tree. So at a typical step of the algorithm, certain subtrees T 1 ; T r of the tree T have already been investigated. Each of these trees T i has a vertex ....

R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In G. Tinhofer, E. Mayr, H. Noltemeier, and M. M. Sys/lo, editors, Computational Graph Theory, Computing Supplementum 7, pages 17--51. Springer-Verlag Wien, 1990.

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R.H. Mohring, Graph problems related to gate matrix layout and PLA folding. Computational graph theory, Comput. Suppl. 7, 17--51 (1990).

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R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In Computational graph theory, volume 7 of Comput. Suppl., pages 17--51. Springer, Vienna, 1990.

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R. H. Mohring. Graph problems related to gate matrix layout and PLA folding. In E. Mayr, H. Noltemeier, and M. Sys#lo, editors, Computational Graph Theory, Computing Suppl. 7, pages 17--51. Springer Verlag, 1990.

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