J. A. Ellis, I. H. Sudborough, and J. S. Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50--79, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 minors developed by Robertson Seymour [3, 7, 22] Furthermore, some search problems have applications in ....

J. A. Ellis, I. H. Sudborough, and J. Turner, The vertex separation and search number of a graph, Information and Computation, 113 (1994), pp. 50-79.

....Garey, Johnson and Papadimitriou [24] proved that determining whether s(G) k is NP complete. They gave an O(n) time algorithm to determine the search number of n node trees, and an O(n log n) time algorithm to determine a minimal search strategy in n node trees. Ellis, Sudborough and Turner [9] linked s(G) with the vertex separation vs(G) of G (known to be equal to the pathwidth of G [17] Ellis et al. showed that vs(G) s(G) vs(G) 2, and that s(G) vs(G ) where G is the 2 augmentation of G, i.e. the network obtained from G by replacing every link fx; yg by a path fx; a; ....

....extremal trees, i.e. trees requiring the maximum number of searchers to be cleared. We show that n node extremal trees require roughly log 2 n searchers to be cleared. This contrasts with the standard edge search problem for which n node extremal trees require only log 3 n searchers to be cleared [9, 24]. 2. DEFINITIONS As for the edge search problem, we are given a contaminated network G = V; E) and the problem consists to obtain, via a sequence of operations using searchers, a state of the network in which all links are simultaneously clear. Again, we assume that G is connected. A search ....

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J. Ellis, H. Sudborough, and J. Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50-79, 1994.

....B i such that both u; v 2 B i ; and 3. for all 1 i j k p, B i B k B j . The width of B is maxfjB i j j 1 i pg 1. The pathwidth of a graph G, denoted pw(G) is the minimum width of a path decomposition of G. Linear algorithms for computing the pathwidth of trees are described in [15, 7, 16]. The next two results about trees and pathwidth are given in [15, 7] Lemma 4 [15, 7] A tree T has pathwidth at most h if and only if for all vertices v in T at most two components of T n v have pathwidth h and the remainder have pathwidth at most h 1. As de ned in [7] we say that a vertex v ....

....B k B j . The width of B is maxfjB i j j 1 i pg 1. The pathwidth of a graph G, denoted pw(G) is the minimum width of a path decomposition of G. Linear algorithms for computing the pathwidth of trees are described in [15, 7, 16] The next two results about trees and pathwidth are given in [15, 7]: Lemma 4 [15, 7] A tree T has pathwidth at most h if and only if for all vertices v in T at most two components of T n v have pathwidth h and the remainder have pathwidth at most h 1. As de ned in [7] we say that a vertex v is h critical in a rooted tree T if exactly two subtrees rooted at ....

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John A. Ellis, Ivan Hal Sudborough, and Jonathan Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50-79, 1994.

....such that both u, v B i ; and 3. for all 1 i j k p, B i B k B j . The width of B is max B i p 1. The pathwidth of a graph G, denoted pw(G) is the minimum width of a path decomposition of G. Linear algorithms for computing the pathwidth of trees are described in [15, 7, 16]. The next two results about trees and pathwidth are given in [15, 7] Lemma 4 [15, 7] A tree T has pathwidth at most h if and only if for all vertices v in T at most two components of T v have pathwidth h and the remainder have pathwidth at most 1. As defined in [7] we say that a vertex ....

....k B j . The width of B is max B i p 1. The pathwidth of a graph G, denoted pw(G) is the minimum width of a path decomposition of G. Linear algorithms for computing the pathwidth of trees are described in [15, 7, 16] The next two results about trees and pathwidth are given in [15, 7]: Lemma 4 [15, 7] A tree T has pathwidth at most h if and only if for all vertices v in T at most two components of T v have pathwidth h and the remainder have pathwidth at most 1. As defined in [7] we say that a vertex v is h critical in a rooted tree T if exactly two subtrees rooted at ....

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John A. Ellis, Ivan Hal Sudborough, and Jonathan Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50--79, 1994. 3, 18 21

....of edge searching and was introduced by Kirousis and Papadimitriou in [14] Finally, mixed searching was introduced in [24] and [2] and is a natural generalisation of the two previous variants (for the formal definitions see Subsection 4. 1 for other results concerning search games on graphs see [1, 7, 9, 15, 16, 23]. The problems of computing es(G) ns(G) ms(G) or linear width(G) is NP complete (see [16, 14, 24, 25] On the other hand, since all of these parameters are closed under taking of minors, we know (see e.g. 3, 19, 20, 22, 21] that, for any k, there exists a linear algorithm that given a ....

J. A. Ellis, I. H. Sudborough, and J. Turner. The vertex separation and search number of a graph. Information and Computation, 113:50--79, 1994.

.... 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 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. ....

....by Bienstock and Seymour [4] and use it in a novel way; in fact, we employ it not to prove monotonicity, but to transform a contiguous strategy into a monotone internal one with the same number of searchers. We also adapt to contiguous search the techniques used by Ellis, Sudborough and Turner [12] for linking search numbers and vertex separation. We then prove a strong di erence between traditional search and both contiguous and monotone internal searches. In fact, we show that, in trees, there is only one obstruction for monotone internal search, as well as for contiguous search. This ....

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J. Ellis, H. Sudborough, and J. Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50-79, 1994.

....k is a fixed constant. See e.g. 11] for an overview. This paper addresses the case that k is a fixed constant. The first known algorithms, solving the treewidth and pathwidth problems for fixed k are based on dynamic programming and use respectively O(n k 2 ) and O(n 2k 4k 8 ) time [3, 21]. Then, Robertson and Seymour [41] gave a non constructive proof of the existence of O(n ) decision algorithms for the problems. Their algorithms consist of two steps. The first step either decides that the treewidth of the input graph G is too large, or finds a tree decomposition of G 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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J. A. Ellis, I. H. Sudborough, and J. Turner. The vertex separation and search number of a graph. To appear in Information and Computation, 1994.

.... (because, with sum, only graphs with very small connected components could be built) To show that the sum operation cannot be dropped from Theorem 7, we now discuss the close relationship between the concatenation operation and the notion of pathwidth (introduced in [RobSey] see also, e.g. [Bod, Klo, EllST]) In the following definition we (slightly) generalize the notion of pathwidth, to (hyper)graphs with begin and end nodes (cf. Cou3] Definition 8. A path decomposition of a graph g is a sequence (V 1 , V n ) 1, of subsets of V g such that (1) # i=1 V i = V g , 2) for every e E g ....

....as required. Theorem22. Int(CF) HR. Proof. Inclusion follows immediately from Theorems 21 and 11. Proper inclusion is a consequence of Corollary 17: the set of all trees is in HR, but is not of bounded pathwidth, as can easily be seen (for a characterization of the trees of pathwidth k, see [EllST]) Since REG is closed under intersection, the proof of Theorem 21 also works for Int(REG) In fact, the proof preserves the right linearity of the grammars. Hence, Int(REG) Val(RLIN CFG(CS) As an example, the graph language of clothes lines is generated by the right linear context free ....

J.A.Ellis, I.H.Sudborough, J.S.Turner; The vertex separation and search number of a graph, Inform. and Comput. 113 (1994), 50-79

.... O(n) 78] Square meshes O(n) 74] Hypercubes O(n) 49] de Bruijn graph of order 4 O(n) 51] d dimensional c ary cliques O(n) 80] Cutwidth Trees O(n log n) 100] Complete k level t ary tree O(1) 68] Hypercubes O(n) 49] d dimensional c ary cliques O(n) 80] VertSep Trees O(n log n) [31] SumCut Trees O(n 1.722 ) 64] Trees O(n) 24] EdgeBis Hypercubes O(n) 80] d dimensional c ary arrays O(n) 80] d dimensional c ary cliques O(n) 80] Table 4: Review of classes of graphs optimally solvable in polynomial time. n denotes the number of nodes in the graph, m the number of ....

J. Ellis, I. H. Sudborough, and J. Turner. The vertex separation and search number of a graph. Information and Computation, (113):50--79, 1979.

.... = lw(G n ) We mention that the mixed search number is equivalent to the parameter of properpathwidth de ned by Takahashi, Ueno, and Kajitani in [22, 23] It is also known that the node search number is equivalent to the pathwidth, the interval thickness, and the vertex separation number (see [8, 11, 12, 13, 18]) 13 Theorem 7 gives a way to transform any searching game to a conquest game for linear width. Therefore, the obstruction characterization for linear width provided by Theorem 3, can serve as an obstruction characterization for any variant of the search parameters. Applying Theorem 3 and ....

J. A. Ellis, I. H. Sudborough, and J. Turner, The vertex separation and search number of a graph, Information and Computation, 113 (1994), pp. 50-79. 15

....separation problem has the same formulation as the minimum cut arrangement problem, but using as measure the number of vertices in the first partition connected to the second one. This measure was first introduced in [7] as the # operator. The problem is NP complete [24] but in P for trees [12]. The global version in which one looks for a layout minimizing the sum of all the separations is known as the minimal sum cut problem [10] or the minimal profile problem [22] The problem is equivalent to the interval graph completion problem that is also NP complete [15] For trees the problem ....

J. Ellis, I. H. Sudborough, and J. Turner. The vertex separation and search number of a graph. Information and Computation, (113):50--79, 1979.

....[33] NP C for trees # = 3 [29] APX for dense graphs [46] caterpillars with hair length 3 [55] no PTAS for trees [9] no APX in general [45] Cutwidth NP C [32] P for trees [75] NP C for pl. graphs # = 3 [57] NC for trees [19] APX for dense graphs [5] VertSep NP C [52] P for trees [28] EdgeBis NP C [31] P for trees (ref. in [15] P for grid graphs without holes [64] PTAS for planar graphs [15] NCAS for planar graphs [26] P Solvable in polynomial time. APX Approximation algorithm in polynomial time. NP Solvable in non deterministic polynomial time. PTAS Has an ....

....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 to VertSep restricted to grid graphs using the same transformation that we used for Cutwidth. The resulting graph H is a grid graph homeomorphic to the input graph G, so we get minvs(H) # K # minsn(G) # K. We ....

J. Ellis, I. H. Sudborough, and J. Turner. The vertex separation and search number of a graph. Information and Computation, (113):50--79, 1979.

....1 graph with multiplicative factor two. Our algorithm is based on structural results on the relation between the pathwidth of a 2 connected outerplanar graph and its dual, which are interesting in their own right. This dual relation combining with the results of Ellis et al. [4] that the pathwidth of trees can be computed in linear time are the main ingredients of our algorithm. Also, we show how to construct the corresponding path decomposition in O(n log n) time. In [5] Govindran et al. give an O(n log n) time algorithm for approximating the pathwidth of an ....

....) Applying Lemma 9 for H we have that pw(H # ) # 2 pw(H) So we have: pw(G) # pw(G # ) # pw(H # ) # 2 pw(H) # 2 pw(G) 2. The lemmas needed for the proof above will follow in the remainder of this section. We need the following fact about pathwidth of trees. Theorem 2 (Ellis et al. [4]) 1. Every tree T of pathwidth k 1, k # 1, has a vertex u such that the forest T u has at least three connected components of pathwidth # k; 3 2. For any tree T , pw(T ) # k 1, k # 1, if and only if there is a path P such that every connected component of the forest T V (P ) ....

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J. A. Ellis, I. H. Sudborough, and J. Turner, The vertex separation and search number of a graph, Information and Computation, 113 (1994), pp. 50--79.

....outerplanar graph with multiplicative factor two. Our algorithm is based on structural results on the relation between the pathwidth of a 2 connected outerplanar graph and its dual, which are interesting in their own right. This dual relation combining with the results of Ellis et al. [4] that the pathwidth of trees can be computed in linear time are the main ingredients of our algorithm. Also, we show how to construct the corresponding path decomposition in O(n log n) time. In [5] Govindran et al. give an O(n log n) time algorithm for approximating the pathwidth of an ....

.... ) Applying Lemma 9 for H we have that pw(H ) 2 pw(H) So we have: pw(G) pw(G ) pw(H ) 2 pw(H) 2 pw(G) 2: The lemmas needed for the proof above will follow in the remainder of this section. We need the following fact about pathwidth of trees. Theorem 2 (Ellis et al. [4]) 1. Every tree T of pathwidth k 1, k 1, has a vertex u such that the forest T n fug has at least three connected components of pathwidth k; 2. For any tree T , pw(T ) k 1, k 1, if and only if there is a path P such that every connected component of the forest T n V (P ) has pathwidth ....

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J. A. Ellis, I. H. Sudborough, and J. Turner, The vertex separation and search number of a graph, Information and Computation, 113 (1994), pp. 50-79.

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J. A. Ellis, I. H. Sudborough, and J. S. Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50--79, 1994.

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J. A. Ellis, Ivan Hal Sudborough, and J. S. Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50--79, 1994.

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J. A. Ellis, I. H. Sudborough, and J. S. Turner. The vertex separation and search number of a graph. Inform. and Comput., 113(1):50--79, 1994.

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J. A. Ellis, I. H. Sudborough, and J. Turner. The vertex separation and search number of a graph. Information and Computation, 113:50--79, 1994.

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J. A. Ellis, I. H. Sudborough, and J. S. Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50--79, 1994.

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J. Ellis, H. Sudborough, and J. Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50--79, 1994.

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J. A. Ellis, I. H. Sudborough, and J. S. Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50--79, 1994.

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J. A. Ellis, I. H. Sudborough, and J. S. Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50--79, 1994.

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J. Ellis, I. Sudborough, J. Turner, The vertex separation and search number of a graph, Inform. Comput. 113 (1994) 50-79.

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J. A. Ellis, I. H. Sudborough, and J. Turner, The vertex separation and search number of a graph, Information and Computation, 113 (1994), pp. 50--79.

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J. A. Ellis, I. H. Sudborough, and J. Turner, The vertex separation and search number of a graph, Information and Computation, 113 (1994), pp. 50--79.

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