F. S. Makedon and I. H. Sudborough. On minimizing width in linear layouts. Disc. Appl. Math., 23:243--265, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....unrelated elds. There is a strong resemblance of graph searching to certain pebble games [15] that model sequential computation. Other applications of graph searching can be found in VLSI theory since this gametheoretic approach to some important parameters of graph layouts such as the cutwidth [19], the topological bandwidth [18] the bandwidth [9] the pro le [10] and the vertex separation number [8] is very useful for the design of ecient algorithms. There is also a connection between graph searching, pathwidth and treewidth, parameters that play an important role in the theory of graph ....

F. S. Makedon and I. H. Sudborough, On minimizing width in linear layouts, Disc. Appl. Math., 23 (1989), pp. 201-298.

....for this success is that the problem and its several variants (node search, mixedsearch, t search, etc. is closely related to standard graph parameters and concepts, including tree width, cut width, path width, and, last but not least, graph minors [4] For instance, Makedon and Sudborough [23] showed that s(G) is equal to the cutwidth of G for all networks of maximum degree 3. Similarly, Kirousis and Papadimitriou showed that the node search number of a network is equal to its intervalwidth [18] and to its vertex separator plus one [19] Seymour and Thomas [28] showed that the ....

F. Makedon and H. Sudborough. Minimizing width in linear layout. In 10th Int. Colloquium on Automata, Languages, and Programming (ICALP '83), LNCS 154, Springer-Verlag, 478-490, 1983.

....all the linear orderings of the vertices or the edges of a graph (for a survey, see [21] One of the most known problems of this type is the problem to compute the cutwidth of a graph. It is also known as the MINIMUM CUT LINEAR ARRANGEMENT problem and has several applications such as VLSI design [2, 3, 35, 33], network relia bility [30] automatic graph drawing [44, 38] and information retrieval [15] Cutwidth has been extensively examined [17, 24, 25, 31, 35, 37, 49] and it appears to be closely related with other graph parameters like pathwidth, linear width, bandwidth, and modi fied bandwidth ....

....to compute the cutwidth of a graph. It is also known as the MINIMUM CUT LINEAR ARRANGEMENT problem and has several applications such as VLSI design [2, 3, 35, 33] network relia bility [30] automatic graph drawing [44, 38] and information retrieval [15] Cutwidth has been extensively examined [17, 24, 25, 31, 35, 37, 49] and it appears to be closely related with other graph parameters like pathwidth, linear width, bandwidth, and modi fied bandwidth [17, 18, 31, 34, 35] Briefly, the cutwidth of a graph G = IV(G)l, IE(G)l) is equal to the nfinimum k for which there exists a vertex ordering of G such that for any ....

[Article contains additional citation context not shown here]

F. S. Makedon and I. H. Sudborough. On minimizing width in linear layouts. Discrete Appl. Math., 23:243 265, 1989.

....the problem and its variants, i.e. node search, mixed search, inert search, etc. are closely related to standard graph parameters and concepts, including treewidth, cutwidth, pathwidth, and linearwidth [2] For instance, s(G) is equal to the cutwidth of G for all graphs of maximum degree 3 (see [25]) Similarly, the node search number of a graph is equal to its pathwidth plus one, and also to its vertex separator plus one [18, 19, 20] The inert search number is equal to the treewidth plus one [10, 32] and the mixed search number is equal to the proper pathwidth [38, 39] For more on graph ....

.... on graph searching, we refer the reader to, e.g. 9, 12, 14, 15] Graph searching is a non trivial interesting and challenging problem; even determining whether s(G) k for arbitrary G and k, is NP complete [26] Not surprisingly, the research has focused on restricted classes of graphs (e.g. [19, 25, 27, 33, 34]) and on bounded search numbers (e.g. Departament de Matem atica Aplicada IV, Universitat Polit ecnica de Catalunya, Spain. lali mat.upc.es. CNRS, Laboratoire de Recherche en Informatique, Universit e Paris Sud, France. http: www.lri.fr pierre. School of Computer Science, Carleton ....

F. Makedon and H. Sudborough. Minimizing width in linear layout. In 10th Int. Colloquium on Automata, Languages, and Programming (ICALP '83), LNCS 154, Springer-Verlag, 478-490, 1983.

....storage time product minimization problem [91] MinLA has also received some alternative names, as Optimal Linear Ordering, Edge Sum problem or Minimum 1 sum. The Cutwidth problem was first used as a theoretical model for the number of channels in an optimal layout of a circuit in the seventies [3, 72]. In general, the cutwidth of a graph times the order of the graph gives a measure of the area needed to represent the graph in a VLSI layout when nodes are laid out in a row [68] More recent applications of this problem include network reliability [59] automatic graph drawing [79] information ....

F. Makedon and I. H. Sudborough. On minimizing width in linear layouts. Discrete Applied Mathematics, 23(3):243--265, 1989.

....the conquest game framework introduced in Section 2 can provide a min max theorem. 4.1 De nitions We give rst general de nitions of some well known width type parameters for graphs. The cutwidth of graphs has been extensively considered and emerged as a tool for the study of VLSI layouts (See [1, 16] for further references) We give below a natural generalization of its de nition. Let H be a graph with the vertex set V (H) and the edge set E(H) For X V (H) let 0 (X) be the number of edges incident to vertices in X and V (H) X. Let = B 1 ; B 2 ; B n ) max 1 i n jB i j m, be ....

F. S. Makedon and I. H. Sudborough, On minimizing width in linear layouts, Discrete Appl. Math., 23 (1989), pp. 243-265.

....all the linear orderings of the vertices or the edges of a graph (for a survey, see [20] One of the most known problems of this type is the problem to compute the cutwidth of a graph. It is also known as the Minimum Cut Linear Arrangement problem and has several applications such as VLSI design [2, 3, 33, 31], network reliability [28] automatic graph drawing [42, 36] and information retrieval [14] Cutwidth has been extensively examined [16, 22, 23, 29, 33, 35, 49] and it appears to be closely related with other graph parameters like pathwidth, linear width, bandwidth, and modi ed bandwidth [16, 17, ....

....to compute the cutwidth of a graph. It is also known as the Minimum Cut Linear Arrangement problem and has several applications such as VLSI design [2, 3, 33, 31] network reliability [28] automatic graph drawing [42, 36] and information retrieval [14] Cutwidth has been extensively examined [16, 22, 23, 29, 33, 35, 49] and it appears to be closely related with other graph parameters like pathwidth, linear width, bandwidth, and modi ed bandwidth [16, 17, 29, 32, 33] Brie y, the cutwidth of a graph G = V (G) E(G) is equal to the minimum k for which there exists a vertex ordering of G such that for any gap ....

[Article contains additional citation context not shown here]

F. S. Makedon and I. H. Sudborough. On minimizing width in linear layouts. Discrete Appl. Math., 23:243-265, 1989.

....The cutwidth of a graph is the minimum cutwidth over all possible layouts of G. Deciding whether, for a given G and an integer k, cutwidth(G) k, is an NP complete problem known in the bibliography as the Minimum Cut Linear Arrangement (see [12] Cutwidth has been extensively examined (see [7, 10, 11, 14, 16, 17, 21]) Supported by the EU project ALCOM FT (IST 99 14186) The research of the rst author was supported by the Ministry of Education and Culture of Spain, Grant number MEC DGES SB98 0K148809. It is closely related with other graph theoretic parameters like pathwidth, bandwidth, modi ed ....

....by the EU project ALCOM FT (IST 99 14186) The research of the rst author was supported by the Ministry of Education and Culture of Spain, Grant number MEC DGES SB98 0K148809. It is closely related with other graph theoretic parameters like pathwidth, bandwidth, modi ed bandwidth (see [15, 7, 16, 14, 8]) and it is approximable (see [9] within a factor of O(log n log log n) in polynomial time (where n is the number of vertices of the input graph) Finally, while it remains NP complete even for planar graphs with maximum degree 3 (see [17] it is polynomially computable for trees (see [21] ....

[Article contains additional citation context not shown here]

F. S. Makedon and I. H. Sudborough. On minimizing width in linear layouts. Disc. Appl. Math., 23:243-265, 1989.

....graphs, all these problems are NP hard. Moreover, the decisional version of Cutwidth and V ertex Separation problems are NP complete even when restricted to lattice graphs and unit disk graphs [10] All of them have a long history, owing to their practical relevance in di#erent applications [35, 15, 1, 32, 24, 23, 2, 26, 7, 17, 19, 32, 18, 5, 22, 27, 33, 34]. Our layout problems are formally defined as follows. A layout # of a graph G = V, E) is a one to one function # : V # 1, n with n = V . Given a graph G, a layout # of G and an integer i, let us define the sets L(i, #, G) u # V (G) #(u) # i and R(i, #, G) u # V ....

F. Makedon and I. H. Sudborough. Minimizing width in linear layouts. In J. Daz, editor, Proc. 10th. International Colloquium on Automata, Languages and Programming, pages 478--490. Springer-V erlag, Lectures Notes in Computer Science, 1983.

.... the Edgesum problem, is relevant in circuit and VLSI layout [72, 35] single machine job scheduling [1, 70] and as a simplified model for nervous system simulation [54] The Minimum Cut Width problem was first used as a theoretical model for the number of channels in an optimal layout of a circuit [53, 2]. More recent applications of the problem include network reliability [42] automatic graph drawing [60] information retrieval [12] and as a subroutine for the cutting plane algorithm to solve the TSP [40] The Minimum Sum Cut problem, also known as the Profile problem, is equivalent to the ....

....G, therefore we have that mincw(G) # K # mincw(H) # K, which proves the claimed result. Observe that the previous reduction creates graphs with maximum degree 3 and recall that for graphs with maximum degree 3, the SearchNb problem (whose measure is sn) is identical to the Cutwidth problem [53]. Therefore, we get as corollary that SearchNb remains NP complete even when restricted to grid graphs. For any graph G, the vertex separation of a homeomorphic image of G is identical to the search number of G [28] Let us reduce SearchNb restricted to planar graphs with maximum vertex degree 3 ....

F. Makedon and I. H. Sudborough. Minimizing width in linear layouts. In J. Daz, editor, Proc. 10th. International Colloquium on Automata, Languages and Programming, pages 478--490. Springer-Verlag, Lectures Notes in Computer Science, 1983.

....The cutwidth of a graph is the minimum cutwidth over all possible layouts of G. Deciding whether, for a given G and an integer k, cutwidth(G) k, is an NP complete problem known in the bibliography as the Minimum Cut Linear Arrangement (see [12] Cutwidth has been extensively examined (see [7, 10, 11, 14, 16, 17, 21]) It is closely related with other graph theoretic parameters like pathwidth, bandwidth, modified bandwidth (see [15, 7, 16, 14, 8] and it is approximable (see [9] within a factor of O(log n log log n) in polynomial time (where n is the number of vertices of the input graph) Finally, while it ....

.... k, is an NP complete problem known in the bibliography as the Minimum Cut Linear Arrangement (see [12] Cutwidth has been extensively examined (see [7, 10, 11, 14, 16, 17, 21] It is closely related with other graph theoretic parameters like pathwidth, bandwidth, modified bandwidth (see [15, 7, 16, 14, 8]) and it is approximable (see [9] within a factor of O(log n log log n) in polynomial time (where n is the number of vertices of the input graph) Finally, while it remains NP complete even for planar graphs with maximum degree 3 (see [17] it is polynomially computable for trees (see [21] ....

[Article contains additional citation context not shown here]

F. S. Makedon and I. H. Sudborough. On minimizing width in linear layouts. Disc. Appl. Math., 23:243-- 265, 1989.

.... and evader can move from vertex to vertex until eventually a pursuer and evader lie in the same vertex [17, 20] The search number of a graph refers to the minimum number of pursuers needed to solve a pursuit evasion problem, and has been closely related to other graph properties such as cutwidth [16, 18]. It has also been shown that a graph can be searched monotonically (i.e. without recontamination) in [2, 12] Pursuit evasion scenarios in continuous spaces have arisen in a variety of applications 1 such as air traffic control [1] military strategy [11] and trajectory tracking [10] This has ....

....: Let the vertices of G k correspond to the set of all points with integer coordinates, i; j) such that 10 0 i; j k. Let G k connect vertices v and w by an edge if and only if v adn w are distance 1 apart (i.e. a standard four neighborhood) The cutwidth of G k is k. It is established in [16] that for all graphs G, the search number S(G) is related by S(G) CW (G) bdeg(G) 2c Delta S(G) in which deg(G) is the maximum vertex degree of G. Because deg(G k ) 4, S(G) k 2S(G) Using Lemma 1, geometric instances of G k such as the one shown in Figure 4 can be constructed. Both G k and ....

F. Makedon and I. H. Sudborough. Minimizing width in linear layouts. In Proc. 10th ICALP, Lecture Notes in Computer Science 154, pages 478--490. Springer-Verlag, 1983.

....the resemblance of graph searching to certain pebble games [18] that model sequential computation. The second motivation of the interest to the graph searching arises from the VLSI theory. Exploitation of gametheoretic approaches to some important parameters of graphs layouts such as the cutwidth [23], the topological bandwidth [22] and the vertex separation number [11] is very useful for the construction of efficient algorithms. Yet another reason is connections between graph searching, the pathwidth and the treewidth. These parameters play very important role in the theory of graph minors ....

....; i k , then 1 2 : k = 1 2 k(k 1) n Gamma1 X i=1 j S i (G; f)j 1 2 : k (n Gamma k)k = 1 2 (2n 1 Gamma k)k. The proof of the second inequality is similar. Note that for any i 6= i 1 ; i 2 ; i k , 1 j S i (G; f)j k. 2 Makedon and Sudborough [23] obtain some bounds for the cutwidth in terms of the maximal degree and the search number. By virtue of (9) it is not surprising that there are strong connections between the bandwidth sum and the search cost. Proposition 9 For any linear layout f of G bw sum (G; f) Delta(G)vs sum (G; f) ....

F. S. Makedon and I. H. Sudborough, On minimizing width in linear layouts, Disc. Appl. Math., 23 (1989), pp. 243--265.

....Initially, problems of guaranteed search on graphs were stated by Parsons in [4] and by Petrov in [5] Since then such problems attracted the attention of many researchers because of their connections with different seemingly unrelated topics. Mention just some of them: ffl linear graph layouts [6] ffl problems of the fight against damage spread in complex systems, for instance, spread of the mobile computer virus in networks [7] ffl pebbling games [8] ffl the theory of graphs minors [9] ffl problems of privacy in distributed systems [10] See surveys [7] and [11] for references. ....

....jf(u; v) 2 E Gamma: f(u) i; f(v) igj: The cutwidth of Gamma with respect to an ordering f is cw( Gamma; f) max i21;jV Gammaj cw i ( Gamma; f) and the cutwidth of Gamma (we denote it by cw( Gamma) is the minimum cutwidth over all linear orderings of Gamma. Makedon and Sudborough in [6] proved Theorem 2.4 For any graph Gamma with the maximal degree 3, S 0 1 ( Gamma) cw( Gamma) Let Gamma be a graph. Denote by Gamma 0 (v) the subgraph of Gamma induced by the set of all vertices adjacent to v. Let n k ( Gamma; v) be the number of connected components of Gamma 0 (v) ....

F. S. Makedon and I. H. Sudborough, On minimizing width in linear layouts, Disc. Appl. Math., 23 (1989), pp. 201--298.

....MEC DGES SB98 0K148809. y This paper is the full version of part of the paper titled Constructive linear time algorithms for small cutwidth and carving width which appeared in the proceedings of ISAAC 2000. 1 mum Cut Linear Arrangement (see [12] Cutwidth has been extensively examined (see [7, 10, 11, 13, 15, 16, 19]) It is closely related with other graph theoretic parameters like pathwidth, bandwidth, modified bandwidth (see [14, 7, 15, 13, 8] and it is approximable within a factor of O(logn log log n) in polynomial time (see [9] Finally, while it remains NP complete even for planar graphs with maximum ....

....and carving width which appeared in the proceedings of ISAAC 2000. 1 mum Cut Linear Arrangement (see [12] Cutwidth has been extensively examined (see [7, 10, 11, 13, 15, 16, 19] It is closely related with other graph theoretic parameters like pathwidth, bandwidth, modified bandwidth (see [14, 7, 15, 13, 8]) and it is approximable within a factor of O(logn log log n) in polynomial time (see [9] Finally, while it remains NP complete even for planar graphs with maximum degree 3 (see [16] it is polynomially computable for trees (see [19] Our results concern the fixed parameter tractability of ....

[Article contains additional citation context not shown here]

F. S. Makedon and I. H. Sudborough. On minimizing width in linear layouts. Disc. Appl. Math., 23:243--265, 1989.

.... E j ) is counted once. PATHWIDTH OF PLANAR AND LINE GRAPHS 7 Finally, j jv i 1 V i j j je j 1 E j j; which completes the proof. 4. LINE GRAPHS OF SMALL DEGREE GRAPHS In [8] Golovach have obtained the following result about the vertex separation of line graphs and cutwidth (see [13] for de nitions and further results on cutwidth) Theorem 4.1 (Golovach) For any graph G, cw(G) vs(L(G) cw(G) b (G) 2c 1; where cw(G) is the cutwidth of G. The well known result of Makedon and Sudborough [13] is that for any graph G of (G) 3 the cutwidth of G is equal to the edge ....

....following result about the vertex separation of line graphs and cutwidth (see [13] for de nitions and further results on cutwidth) Theorem 4.1 (Golovach) For any graph G, cw(G) vs(L(G) cw(G) b (G) 2c 1; where cw(G) is the cutwidth of G. The well known result of Makedon and Sudborough [13] is that for any graph G of (G) 3 the cutwidth of G is equal to the edge search number. Since the edge search number of G is at most vs(G) 2, we obtain the following corollary of Golovach s theorem. We refer the reader to [1] for a survey on graph searching. Lemma 4.1. For any graph G of ....

F. S. Makedon and I. H. Sudborough, On minimizing width in linear layouts, Discrete Appl. Math., 23 (1989), pp. 243-265.

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F. S. Makedon and I. H. Sudborough. On minimizing width in linear layouts. Disc. Appl. Math., 23:243--265, 1989.

No context found.

F. Makedon and I. H. Sudborough. Minimizing width in linear layouts. In Proc. 10th ICALP, Lecture Notes in Computer Science 154, pages 478--490. Springer-Verlag, 1983.

No context found.

F. S. Makedon and I. H. Sudborough, On minimizing width in linear layouts, Discrete Appl. Math., 23:243--265, 1989.

No context found.

F. S. Makedon and I. H. Sudborough. On minimizing width in linear layouts. Discrete Appl. Math., 23:243--265, 1989.

No context found.

F. S. Makedon and I. H. Sudborough. On minimizing width in linear layouts. Disc. Appl. Math., 23:243--265, 1989.

No context found.

F. Makedon and H. Sudborough. Minimizing width in linear layout. In 10th Int. Colloquium on Automata, Languages, and Programming (ICALP '83), LNCS 154, Springer-Verlag, 478--490, 1983.

No context found.

F. S. Makedon and I. H. Sudborough, On minimizing width in linear layouts, Disc. Appl. Math., 23 (1989), pp. 201--298.

No context found.

F. S. Makedon and I. H. Sudborough, On minimizing width in linear layouts, Disc. Appl. Math., 23 (1989), pp. 201--298.

No context found.

F. Makedon and I.H. Sudborough, Minimizing width in linear layouts, in: Proc. 10th ICALP, Lecture Notes in Computer Science 154, Springer-Verlag (1983) 478-490.

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