L.M. Kirousis and C.H. Papadimitriou. Searching and pebbling. Theor. Comput. Sci. 47, 205--218 (1986).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....versions of graph searching has been attracting the attention of researchers from Discrete Mathematics and Computer Science for a variety of elegant and unexpected applications in di erent and seemingly 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 ....

.... problems have applications in problems of privacy in distributed environments with mobile eavesdroppers ( bugs ) 11] In the standard node search version of searching, a single searcher is placed at a vertex of a graph G at every move, while from other vertices searchers are removed (see e.g. [15]) The purpose of searching is to capture an invisible fugitive moving fast along paths in G. The fugitive is not allowed to run through the vertices currently occupied by searchers. So the fugitive is caught when a searcher is placed on the vertex it occupies, and it has no possibility to leave ....

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L. M. Kirousis and C. H. Papadimitriou, Searching and pebbling, Theor. Comp. Sc., 47 (1986), pp. 205-218.

....a searcher from a node. The fugitive is captured if both ends of the edge where he hides are simultaneously occupied by a searcher. The fugitive is allowed to move (at any speed) along edges subject to the condition that he never passes a node occupied by a searcher. Kiriousis and Papadimitriou [28] have shown that (G) ns(G) Moreover there is an optimal search, i.e. a search requiring only ns(G) searchers, such that after an edge has been cleared by two searchers simultaneously guarding its end nodes it never recontaminates, i.e. there never appears a path that carries no searcher ....

L.M. Kirousis and C.H. Papadimitriou. Searching and pebbling. Theor. Comput. Sci. 47, 205-218 (1986).

....node, and mixed search number of a graph (namely, es(G) ns(G) and ms(G) The first graph searching game was introduced by Breisch [6] and Parsons [18] and is the one of edge searching. Node searching appeared as a variant 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) ....

....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 graph G checks whether es(G) ns(G) ms(G) or linear width(G) is at most k (in other words, all these ....

[Article contains additional citation context not shown here]

L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theor. Comp. Sc., 47:205--218, 1986.

....node, and mixed search number of a graph (namely, es(G) ns(G) and ms(G) The first graph searching game was introduced by Breisch [9] and Parsons [27] and is the one of edge searching. Node searching appeared as a variant of edge searching and was introduced by Kirousis and Papadimitriou in [22]. Finally, mixed searching was introduced in [35] and [3] and is a natural generalisation of the two previous variants (for the formal definitions see Subsection 5.1) The problem of computing es(G) ns(G) ms(G) or linear width(G) is NP complete (see [24, 22, 35] and Theorem 5.i of this paper) ....

....by Kirousis and Papadimitriou in [22] Finally, mixed searching was introduced in [35] and [3] and is a natural generalisation of the two previous variants (for the formal definitions see Subsection 5. 1) The problem of computing es(G) ns(G) ms(G) or linear width(G) is NP complete (see [24, 22, 35] and Theorem 5.i of this paper) On the other hand, since all of these parameters is closed under taking of minors, we know that there exist a linear 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 ....

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L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theor. Comp. Sc., 47:205--218, 1986.

....search number [P] interval thickness [KF] node search number [KP2] and minimum demands in some types of pebble games , too. Some of those studies were motivated by practical problems in VLSI design and other important applications; for further references, see the recent survey [Mo2] and also [KP1]. 4 Trees So far the only general class of graphs for which narrowness is well characterized is the class of trees (and forests) The importance of the result to be discussed in this section is demonstrated by the fact that its various equivalents and consequences were discovered by many authors ....

L.M. Kirousis and Ch.H. Papadimitriou, Searching and pebbling, Theoret. Computer Sci. 47 (2) (1986) 205-218.

....subject classifications. 05C78, 05C50, 68R10 1 Introduction Search problems on graphs attract the attention of researchers from different fields of Mathematics and Computer Science for a variety of reasons. In the first place, this is 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 ....

.... motion coordinations of multiple robots [30] and in problems of privacy in distributed environments with mobile eavesdroppers ( bugs ) 13] More information on graph searching and related problems one can find in surveys [1, 12, 25] In the classical node search version of searching (see, e.g.[18]) at every move of searching a searcher is placed at a vertex or is removed from a vertex. Initially, all edges are contaminated (uncleared) A contaminated edge is cleared once both its endpoints are occupied by searchers. A clear edge e is recontaminated if there is a path without searchers ....

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L. M. Kirousis and C. H. Papadimitriou, Searching and pebbling, Theor. Comp. Sc., 47 (1986), pp. 205--218.

....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. Further we use the word graph to denote a finite connected topological graph, embedded in R 3 . For simplicity, we shall assume that edges of a graph are ....

L. M. Kirousis and C. H. Papadimitriou, Searching and pebbling, Theor. Comp. Sc., 47 (1986), pp. 205--218.

....speed is less than 2 3 and he cannot overcome the distance l(p) in a time t Gamma t Gamma 3 2 l(p) 2 Corollary 10 For any k 0 there is a graph G such that (G) m (G) k. Proof. Let k be an integer and let G be a tree with pw(G) 3 2 k. For each k such a tree exists (see, e.g. [6]) For any homeomorphic image G 0 of a graph G, pw(G) pw(G 0 ) see, e.g. 8] Since every tree is outerplanar, then by Theorem 8 there is a homeomorphic image G 0 of G such that (G 0 ) 2 3 . G) m (G) 2 3 1 m (G) by (1) 2 3 pw(G) k: 2 Concluding remarks. From Theorem 5 ....

L. M. Kirousis and C. H. Papadimitriou, Searching and pebbling, Theor. Comp. Sc., 47 (1986), pp. 205--218.

....out that the suggested algorithms do not run in O(n) as stated but in O(n log n) to clear G. An edge search strategy of G is optimal if it needs the smallest possible number of searchers (this number is es(G) Node searching is another version of graph searching and has been introduced in [11]. In this slightly different version the third activity of the edge searching disappears. An edge is cleared once both endpoints are simultaneously guarded by searchers. The node search number of G is denoted by ns(G) Interval thickness has been introduced in [10] Given a set of intervals I of ....

....a given optimal layout L 0 of G 0 into an optimal edge search strategy S of G in linear time. Since the size of L 0 and G 0 is O(size(G) we imply that for every graph the computation of its optimal edge search strategy requires no more time than the computation of its optimal layout. In [11] it has been shown that ns(G) vs(G) 1. The authors have also presented an algorithm which (by using a suitable data structure) transforms a given optimal layout into an optimal node search strategy in linear time. In [10] it has been shown that (G) ns(G) and that a given optimal node search ....

L. Kirousis and C. Papadimitriou. Searching and pebbling. Theor. Comp. Science, 47:205-- 218, 1986.

....node, and mixed search number of a graph (namely, es(G) ns(G) and ms(G) The first graph searching game was introduced by Breisch [9] and Parsons [30] and is the one of edge searching. Node searching appeared as a variant of edge searching and was introduced by Kirousis and Papadimitriou in [25]. Finally, mixed searching was introduced in [39] and [3] and is a natural generalization of the two previous variants (for the formal definitions see Subsection 5.1 for analogues versions of the searching game without the agility requirement see [13,36] The problem of computing es(G) ....

....and [3] and is a natural generalization of the two previous variants (for the formal definitions see Subsection 5. 1 for analogues versions of the searching game without the agility requirement see [13,36] The problem of computing es(G) ns(G) ms(G) or linear width(G) is NPcomplete (see [27,25,39] and Theorem 25.i of this paper) On the other hand, since all of these parameters are closed under taking of minors, we know that there exists a linear time 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 ....

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L.M. Kirousis, C.H. Papadimitriou, Searching and pebbling, Theor. Comp. Sc. 47 (1986) 205--218. 38

....the inert case agile edge search is not equivalent to agile mixed search. Edge search was the search game to be defined first, introduced by Breisch [4] and Parsons [18] see also [16] Node 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 ....

.... 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 ....

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L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theor. Comp. Sc., 47:205--218, 1986. 20

....in which all edges are simultaneously clear by a sequence of moves using searchers. The objective is to achieve the desired state by using the least number of searchers. In this paper, two versions of graph searching problem, the edge searching problem [MHGJP88] and the node searching problem [KP86], are discussed. These two problems are different in the ways of how searchers are moved in the graph and how contaminated edges are cleared. In node searching, the allowable moves are (1) placing a searcher on a vertex and (2) removing a searcher from a vertex. In node searching, a contaminated ....

....recontamination of any edges. It was shown that the edge searching problem is NP complete on general graphs and can be solved in linear time on trees [MHGJP88] This problem remains NP complete for planar graphs with maximum vertex degree three [MS88] For node searching, it was also shown in [KP86] and [BS91] that recontamination does not help in node searching a graph with ns(G) searchers. Note that the gate matrix layout problem [Mo90] the interval thickness problem [KP85] the pathwidth problem [RS83] the narrowness problem [KT92] and the vertex separation problem [Ki92] are all ....

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L.M. Kirousis and C.H. Papadimitriou, Searching and pebbling, Theoretical Comput. Scie., 47(1986), 205-218.

.... 1983] Hence the search number problem has practical value, as well as theoretical interest, since finding the cutwidth of a graph is important in some VLSI layout applications [Leiserson 1980] A variation on search number, called node search number, was defined by Kirousis and Papadimitriou [Kirousis 1986]. For node search number, a searcher does not need to move along an edge, looking along it is sufficient to catch the fugitive. The authors showed that node search number is identical to vertex separation 1. Vertex separation is related, directly or indirectly, to several other important graph ....

Kirousis, L. M. and Papadimitriou, C. H. (1986), "Searching and Pebbling", Theoretical Computer Science, 47, pp. 205-216.

.... theory to gate matrix layout is discussed (among other problems) in a series of papers [39] 41] 40] For a survey about computational implications of the Robertson Seymour theory, we refer to [72] 13.3 Node searching This problem formulation refers to a searching game on graphs introduced in [86] as a variant of the more investigated edge searching [141] In node searching, the edges of a graph represent a system of pipes or tunnels that are considered contaminated by a gas. The object of node searching is to clear all edges by a search. A search is a sequence of moves where a player ....

....vertices) from the already searched part (all vertices that carried a searcher in the past) A search is called optimal if the maximum number of searchers on the graph at any point is as small as possible. This number is called the node search number of G, and denoted by ns(G) It was shown in [86] (a simpler proof is given in [10] that there always is an optimal search without recontamination of cleared edges. This was used in [85] to show the following unexpected relationships to interval graph augmentation: Theorem 13.9 (Kirousis Papadimitriou) For any graph G, ns(G) t(G) The ....

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L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theor. Comp. Sc., 47:205--218, 1986.

....edge search is not equivalent to mixed search (see also Fig. 1 in the end of this section) Edge search was the search game to be defined first, introduced by Breisch [4] and Parsons [13] Node search appeared as the first variant of edge search and was introduced by Kirousis and Papadimitriou in [10]. Finally, mixed search was introduced in [2] and [16] 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 [5,6,8 10,12] Finally, ms(G) is equal to the parameter of proper pathwidth introduced in [16] The ....

.... search appeared as the first variant of edge search and was introduced by Kirousis and Papadimitriou in [10] Finally, mixed search was introduced in [2] and [16] 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 [5,6,8 10,12]) Finally, ms(G) is equal to the parameter of proper pathwidth introduced in [16] 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 ....

[Article contains additional citation context not shown here]

L. M. Kirousis and C. H. Papadimitriou, Searching and pebbling, Theor. Comp. Sc. 47 (1986) 205--218.

....set, of an objective function defined on such permutations. These objective functions arise in a number of NP complete problems, including MIN CUT LINEAR ARRANGEMENT, MODIFIED MIN CUT [EST] PATH WIDTH [RS2] GATE MATRIX LAYOUT [DKL] VERTEX SEPARATION [Le] SEARCH NUMBER [Pa] NODE SEARCH NUMBER [KP], TWO DIMENSIONAL GRID LOAD FACTOR [FL3] and others. Each of these is known to possess an O(n 2 ) time decision algorithm for any fixed width value [FL3, FL4] Our strategy yields efficient search algorithms for problems that satisfy the following uniformity condition concerning the complexity ....

M. Kirousis and C. H. Papadimitriou, "Searching and Pebbling," Theoretical Computer Science 47 (1986), 205-218.

....node, and mixed search number of a graph (namely, es(G) ns(G) and ms(G) The first graph searching game was introduced by Breisch [7] and Parsons [19] and is the one of edge searching. Node searching appeared as a variant of edge searching and was introduced by Kirousis and Papadimitriou in [15]. Finally, mixed searching was introduced in [27] 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, 8, 10, 16, 17, 25] The problems of computing es(G) ns(G) ms(G) ....

....in [27] 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, 8, 10, 16, 17, 25] The problems of computing es(G) ns(G) ms(G) or linear width(G) is NP complete (see [17, 15, 27, 29]) On the other hand, since all of these parameters are closed under taking of minors, we know (see e.g. 3, 20, 21, 23, 22] that, for any k, there exists a linear algorithm that given a graph G checks whether es(G) ns(G) ms(G) or linear width(G) is at most k (in other words, all these ....

[Article contains additional citation context not shown here]

L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theor. Comp. Sc., 47:205--218, 1986.

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L.M. Kirousis and C.H. Papadimitriou. Searching and pebbling. Theor. Comput. Sci. 47, 205--218 (1986).

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L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theor. Comp. Sc., 47, 205--218, 1986.

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L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theoret. Comput. Sci., 47(2):205--218, 1986.

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L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theor. Comp. Sc., 47:205--218, 1986. 20

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L. Kirousis, C. Papadimitriou, Searching and pebbling, Theor. Comput. Sci. 47 (1986) 205-218.

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L. M. Kirousis and C. H. Papadimitriou, Searching and pebbling, Theor. Comp. Sc., 47 (1986), pp. 205--218.

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L.M. Kirousis and C.H. Papadimitriou, Searching and pebbling. Theor. Comput. Sci. 47, 205-218 (1986).

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L. M. Kirousis and C. H. Papadimitriou, Searching and pebbling, Theor. Comp. Sc., 47 (1986), pp. 205--218.

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