Megiddo, Hakimi, Garey, Johnson, and Papadimitriou. The complexity searching graph. the January 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....T be a tree. Then pw(T ) max k (T is k strict) pw(T ) 1. Proof. For the proof we compare the strictness of a tree T to its node search number ns(T ) pw(T ) 1 (refer to the beginning of this section for the definitions) In the proof we make use of a lemma attributed to Parsons [32] in [29] [Parsons Lemma] For any tree T and integer k 1, ns(T ) k 1 if and only if T has a vertex v at which there are three or more branches that have search number k or more. First we show that for a tree T k strictness implies ns(T ) k, by induction on k. Assume T is 2 strict. Then T ....

N. Meggido, S. Hakimi, M. Garey, D. Johnson, and C. Papadimitriou. The complexity of searching a graph. JACM 35, 18-44, 1988.

....of search steps that results in all links being simultaneously clear. The search number s(G) of a network G is the smallest number of searchers for which a search strategy exists. A search strategy using s(G) searchers in G is called minimal. Megiddo, Hakimi, 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 ....

....system while insuring mutual protection and or permanent easy and secure access to the entering point. This motivates the study of edge search where the removal of an agent is not allowed. Even just this assumption considerably changes the nature of the problem. For instance, it is mentioned in [24] that there are trees for which minimal search strategies (i.e. search strategies using the minimum number of searchers) without removal require n log n) moves (i.e. link traversal) whereas, if the removal of agents is allowed, then, for any network G, there exists a minimal search strategy ....

[Article contains additional citation context not shown here]

N. Megiddo, S. Hakimi, M. Garey, D. Johnson and C. Papadimitriou. The complexity of searching a graph. Journal of the ACM 35(1):18-44, 1988.

....not known, have been considered in the past. We have the deterministic search games, where a fugitive that possesses some properties (it is agile or inert) hides in the nodes or edges of a graph and the aim is to locate it using as few searchers as possible (for definitions and relevant theory see [5, 14, 18]) Also, the problem of exploring an unknown graph has been considered (see [2, 19, 20] for example) Closer to the spirit of our work are the search problems that are defined and studied in [4] where the authors consider problems of locating points in the plane using incomplete knowledge about ....

N. Megiddo, S. Hakimi, M. Garey, D. Johnson, and C. Papadimitriou, The complexity of searching a graph, Journal of the ACM 35 (1988), 18--44.

.... characteristics that define the search variant (e.g. it may be moving constantly or it may move only when some searcher is about to visit the node at which it resides) hides in the nodes of the graph and the aim is to capture it using the smallest possible number of searchers (see, for example, [6, 10, 15]) Deterministic algorithms that explore graphs with unknown topology have been studied in [3, 17] There also exists a class of games called stochastic games in which the opponents employ move strategies involving a probability transition matrix (see [20] A description of games that involve ....

N. Megiddo, S. Hakimi, M. Garey, D. Johnson, and C. Papadimitriou (1988) The complexity of searching a graph. Journal of the ACM, 35, 18--44.

....the best possible route. Search problems in graphs, where the identity of the node that contains the information sought is not known, have been considered before. These include deterministic search games, where a fugitive that possesses some properties hides in the nodes or edges of a graph [5, 13, 14]) and the problem of exploring an unknown graph [2, 12, 16] Our model is similar in spirit to the model in [4] where the authors propose algorithms to search for a point on a line or on a lattice drawn on the plane. However, in that model, the nodes have limited if any knowledge of where the ....

N. Megiddo, S. Hakimi, M. Garey, D. Johnson, and C. Papadimitriou, The complexity of searching a graph, Journal of the ACM 35 (1988), 18-44.

....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 ....

....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 ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 35:18--44, 1988.

....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 ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 35:18--44, 1988.

.... 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. Departament de Matem atica Aplicada IV, Universitat Polit ecnica de Catalunya, Spain. lali mat.upc.es. CNRS, Laboratoire de Recherche en ....

....after clearing a connected set X of edges, remove the searchers and place them in another part of the graph, usually disconnected from X. The problem of determining minimal search strategies under the contiguity constraint is still NP complete in general (it follows from the reduction in [26], as observed in [1] it has been shown in [1] that minimal contiguous strategies can however be computed in linear time for trees. The next property is perhaps the more practically relevant. A search strategy is internal if, once placed, searchers can only move along the graph edges (i.e. they ....

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N. Megiddo, S. Hakimi, M. Garey, D. Johnson and C. Papadimitriou. The complexity of searching a graph. Journal of the ACM, 35(1):18-44, 1988.

....sequence of placing, removing, or moving a pebble along an edge) such that no edge that is cleared at a point of time can be recontaminated again, i.e. if the fugitive is known not to be in edge e then there is no chance for him to enter edge e again in the remainder of the search. Meggido et al. [11] proved that the computation of the search number of a graph is an NP hard problem which implies its NP completeness because of LaPaugh s result. If an edge can be cleared without moving along it, but it suffices to look into an edge from a vertex, then the minimum number of guards needed to ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. Journal of the ACM, 35(1):18--44, 1988.

....problem considered here. In terms of the restaurant finding problem, routing under faults corresponds to the problem where the exact address of the restaurant is known and one wishes avoid roadblocks or traffic jams. Other related lines of research including searching for a fugitive in a graph [4, 12, 15] or exploring an unknown graph [2, 7, 16] In neither case do the nodes contain information of the type considered here that might direct the search. In the restaurant analogy, there are no informed policeman. We feel our work is closer in spirit to that of computing with uncertainty or with noisy ....

N. Megiddo, S. Hakimi, M. Garey, D. Johnson and C. Papadimitriou, "The complexity of searching a graph," JACM, 35 (1988), 18-44.

.... some characteristics that define the search variant (e.g. it may be moving constantly, or it may move only when some searcher is about to visit the node at which it resides) hides in the nodes of the graph; the aim is to capture it using the smallest possible number of searchers (see, for example, [12, 13, 14]) Deterministic algorithms that explore graphs with unknown topology have been studied in [15, 16] There also exists a class of games called stochastic games in which the opponents employ move strategies involving a probability transition matrix (see [17] A description of games called ....

Megiddo, N., Hakimi, S., Garey, M., Johnson, D. and Papadimitriou, C. (1988) The complexity of searching a graph. J. ACM, 35, 18--44.

.... pathwidth, developed in [20] and [3] were the cornerstones for proving the monotonicity of the corresponding graph searching variants (see also [7] Our paper is motivated and constitutes an extension of the ideas in the proofs of the monotonicity of the agile fugitive search games examined in [12, 13, 17, 22, 2, 4, 3, 15] as well as the proofs of the min max theorems in [21] and [19] Our main observation is that the kernel argument of all these proofs is based on the fact that, in any game variant, the cost of the search can be expressed by a connectivity function that is a nonnegativevalued function on the set ....

....it provides obstruction characterizations, game counterparts, and monotonicity proofs for the parameters of linear width, cutwidth and their extensions. Finally, our general min max theorem implies in a uniform way the monotonicity proofs of all the agile fugitive search games developed so far in [12, 17, 22, 4, 15]. To illustrate the main motivation of our research let us give a simple example of an expansion game. Suppose that we have a set of countries subject to join some organization. At every moment of time we can either add a bounded number of countries to the union or expel an arbitrary number of ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou, The complexity of searching a graph, J. ACM, 35 (1988), pp. 18-44.

....be searchable. Related work There have been many researches on detecting the unpredictable intruder. The general problem was first studied in the context of graphs, where the searchers and the intruder can move from vertex to vertex until a searcher and the intruder eventually lie in one vertex [11, 13]. After adopting geometric free space constraints and visibility of the searchers, this problem has attracted much attention in computational geometry and robotics [1, 3 6, 8, 10, 15, 16, 18] As the first attempt, Suzuki and Yamashita [15] introduced the polygon search problem, which is the ....

N. Megiddo, S.L. Hakimi, M.R. Garey, D.S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. Journal of the ACM, pages 18--44, 1988.

....or to verify that no target is present in the polygon. The guards see a target when there is an unobstructed line of sight between it and one of the guards. We may impose various limitations on the viewing frustum and the range of the vision sensors of the guards. Parsons [24] and Megiddo et al. [22] study a similar problem in the context of pursuit evasion in a graph; in this scenario, the guards and target can move from vertex to vertex of a graph, until a guard and the target eventually lie in the same vertex. In our geometric setting, what makes this problem challenging is the issue of ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. On the complexity of searching a graph. J. ACM, 35:18--44, 1988.

....this paper is an extension or combination of problems that have been considered in several contexts. Interesting results have been obtained for pursuit evasion in a graph, in which the pursuers 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 ....

....worst case problem instances. Let Parsons problem refer to the graph5 e e Figure 2: A corridor of this shape disconnects second order visibility between the two entrances, and can be used to construct geometric equivalents of Parsons problem for planar graphs. searching problem presented in [17, 20]. The task is to specify the number of pursuers required to find an evader that can execute continuous motions along the edges of a graph. Instead of using visibility, capture is achieved when one of the pursuers touches the evader. Let G represent a graph, and S(G) represent the number of ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 35(1):18--44, January 1988.

....Introduction Communication networks are vulnerable to privacy violations. Surveillance of the network is one way to deter eavesdroppers. This gives rise to various models of pursuit and evasion on graphs and corresponding complexity considerations. One problem that has been examined in depth (see [5, 16] and references) is the search of a graph by a team of searchers traversing the edges of the graph in pursuit of a mobile fugitive. The minimum number of searchers necessary to detect the fugitive with certainty is called the search number of the graph. Computing it is easy for trees but NP hard ....

....is the search of a graph by a team of searchers traversing the edges of the graph in pursuit of a mobile fugitive. The minimum number of searchers necessary to detect the fugitive with certainty is called the search number of the graph. Computing it is easy for trees but NP hard for general graphs [16]. Extensions of this approach to models of privacy in distributed environments are studied in [10] We consider a similar, but more static situation: The hider selects an arbitrary node of the graph for hiding and must stay there. The searcher selects a starting node and traverses the graph along ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou (1988), The complexity of searching a graph. Journal of the Association for Computing Machinery 35, 18--44.

....between the visited and the yet undiscovered part of the input graph. In a move, the border of separating edges continuously moves beyond the read subgraph. Thus already cleared nodes and edges cannot be recontaminated. This means a monotone search strategy, graph searching without recontamination [BS91, LaP93, MHGJP88]. Hence, graph automata are plans for monotone search strategies on graphs. The search strategies are special. They are given by a finite set of instructions and can be executed by nondeterministic finite state machines. Our main result states that graph automata are equivalent to linear graph ....

N. Megiddo. S.L. Hakimi, M. R. Garey, D.S. Johnson, C.H. Papadimitriou. The complexity of searching a graph. J. Assoc. Comput. Mach. 35 (1988), 18-44.

....Searching and Interval Completion F.V. Fomin P.A. Golovach y Abstract In the early studies on graph searching ([24, 26, 27, 28]) a graph was considered as a system of tunnels in which a fast and clever fugitive is hidden. The classical search problem is in finding a search plan using the minimal number of searchers. In this paper we consider a new criterion of optimization, namely, the search cost. Firstly we prove ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou, The complexity of searching a graph, J. ACM, 35 (1988), pp. 18--44.

....We present a linear time algorithm for computing linear layouts of trees which are optimal with respect to vertex separation. The best algorithm known so far is given by [7] and needs O(n log n) time. Our result solves several other related open problems on trees as for example the one of [17]. Key words. linear graph algorithms, optimal linear layouts, optimal search strategies, optimal interval augmentations AMS subject classifications. 68R10, 05C05, 05C85 1. Introduction. The vertex separation of graphs has been introduced in [15] in the form of a vertex separator game. It is an ....

....G is possible in O(t(n) time where t(n) is the time required by the construction of an optimal layout of G. Out of the complexity point of view given an input graph G and an integer l the problem whether or not vs(G) l and the problem whether or not es(G) l are both NP complete, see [15] and [17] respectively. As a result of the discussion above we obtain that the problems whether or not ns(G) l, and (G) l are also NP complete. They remain NP complete even for chordal graphs [11] starlike graphs [11] and for planar graphs with maximum degree 3. The last claim is obtained from the ....

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N. Megiddo, S. Hakimi, M. Garey, D. Johnson, and C. Papadimitriou, The complexity of searching a graph, J. Assoc. Comput. Mach., 35 (1988), pp. 18--44.

....or to verify that no target is present in the polygon. The guards see a target when there is an unobstructed line of sight between it and one of the guards. We may impose various limitations on the viewing frustum and the range of the vision sensors of the guards. Parsons [24] and Megiddo et al. [22] study a similar problem in the context of pursuit evasion in a graph; in this scenario, the guards and target can move from vertex to vertex of a graph, until a guard and the target eventually lie in the same vertex. In our geometric setting, what makes this problem challenging is the issue of ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. On the complexity of searching a graph. J. ACM, 35:18-44, 1988.

....As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the running time of former algorithms from O(n log n) to O(n) and answers two related open questions raised in [7] and [15]. 1 1 Introduction The vertex separation of graphs has been introduced in [13] in the form of a vertex separator game. It is an important concept with theoretical and practical value. Its closer relation to the black white pebble game of [6] constitutes the theoretical value in complexity ....

....an optimal layout of G. The computation of an optimal path decomposition is possible in time O(t(n) d) where d is the size of the computed path decomposition. Unfortunately given a graph G and an integer k the problems whether or not vs(G) k and es(G) k are both NP complete, see [13] and [15], respectively. By the discussion above we obtain that the problems whether or not ns(G) k, G) k, and pw(G) k are also NP complete. They remain NP complete even for chordal graphs [8] starlike graphs [8] and for planar graphs with maximum degree three [17, 14] For some special graphs ....

[Article contains additional citation context not shown here]

N. Megiddo, S. Hakimi, M. Garey, D. Johnson, and C. Papadimitriou. The complexity of searching a graph. J. Assoc. Comput. Mach., 35(1):18--44, 1988.

....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 ....

N. Megiddo, S.L. Hakimi, M.R. Garey, D.S. Johnson, C.H. Papadimitriou, The complexity of searching a graph, J. ACM 35 (1988) 18--44.

....is agile, i.e. it always moves, no matter if the search threatens it or not) We stress that, in contrast with 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 ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 35:18--44, 1988.

....actively avoiding detection (e.g. search and capture missions) whereas in other cases their motion is approximately random (e.g. search and rescue operation) The latter problems are often called games against nature. Deterministic pursuit evasion games on finite graphs have been well studied [1, 2]. In these games, the region in which the pursuit takes place is abstracted to be a finite collection of nodes and the allowed motions for the pursuers and evaders are represented by edges 1 This research was supported by the Office of Naval Research (grant N00014 97 1 0946) connecting the ....

....of the search number s(G) of a given graph G. By the search number it is meant the smallest number of pursuers needed to capture a single evader in finite time, regardless of how the evader decides to move. It turns out that determining if s(G) is smaller than a given constant is NPhard [2, 3]. Pursuit evasion games on graphs have been limited to worst case motions of the evaders. When a region in which the pursuit takes place is abstracted to a finite graph, the sensing capabilities of each pursuer becomes restricted to a single node: the node occupied by the pursuer. The question ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. Journal of the ACM, 35(1):18--44, Jan. 1988.

....We would like to obtain a state of the graph 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 ....

....it was shown in [La93] and [BS91] that there always exists an optimal edge search strategy for G with es(G) searchers that does not involve 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 ....

N. Megiddo, S.L. Hakimi, M.R. Garey, D.S. Johnson, and C.H. Papadimitriou, The complexity of searching a graph, J. Assoc. Comput. Mach., 35(1988), 18-44.

....minimum number of searchers. A search strategy that does not recontaminate any edge will be called a progressive search strategy. The search number problem is then clearly in NP since it is easy to see a non deterministic, polynomial time solution to the progressive search problem. Meggido et al. [Megiddo 1988] showed that determining the search number of a graph is NP hard, which implies it is NP complete because of LaPaugh s result. They also showed that the search number of a tree can be determined in linear time. It is known that, for any graph G with maximum vertex degree 3, s (G ) is identical to ....

....Insert Figures 2.1 (a) and (b) here 3. The Vertex Separation of Trees Properties of trees can often be computed recursively and in polynomial time by computing the property for subtrees and combining the results. Meggido et al. [Megiddo 1988] give such an algorithm for computing the search number of a tree and Chung et al. Chung 1982] give such an algorithm for computing the cutwidth of trees of fixed vertex degree, d , in time O (n log d n ) Yannakakis [Yannakakis 1985] gives an O (n log n ) cutwidth algorithm for arbitrary trees ....

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Megiddo, N., Hakimi, S. L., Garey, M. R., Johnson, D. S. and Papadimitriou, C. H. (1988), "The Complexity of Searching a Graph ", JACM, 35, pp. 18-44.

.... both endpoints as in node searching) It is therefore possible [86] to obtain (optimal) node searches on G from (optimal) edge searches on a slight modification of G (replace each edge of G by three parallel edges) Exploitation of this transformation and known results for edge searching on trees [122] and dynamic programming formulations [35] yield a linear time search algorithm for trees (see also Section 14) and a polynomial time algorithm of order O(n 2k 2 4k 8 ) for graphs with search number at most k; k fixed. Finally, for a combination of node searching and edge searching with a ....

....Most of the arguments leading to polynomial algorithms come from node searching (in particular Lemma 13.11 and Lemma 13.12) and demonstrate again the usefulness of this interpretation. We will start with the class of trees. As mentioned before, the polynomial algorithm for edge searching on trees [122] can be 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 ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. Assoc. Comp. Mach., 35:18--44, 1988.

....or to verify that no target is present in the polygon. The guards see a target when there is an unobstructed line of sight between it and one of the guards. We may impose various limitations on the viewing frustum and the range of the vision sensors of the guards. Parsons [24] and Megiddo et al. [22] study a similar problem in the context of pursuit evasion in a graph; in this scenario, the guards and target can move from vertex to vertex of a graph, until a guard and the target eventually lie in the same vertex. In our geometric setting, what makes this problem challenging is the issue of ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. On the complexity of searching a graph. J. ACM, 35:18--44, 1988.

....actively avoiding detection (e.g. search and capture missions) whereas in other cases their motion is approximately random (e.g. search and rescue operation) The latter problems are often called games against nature. Deterministic pursuit evasion games on finite graphs have been well studied [1, 2]. In these games, the region in which the pursuit takes place is abstracted to be a finite collection of nodes and the allowed motions for the pursuers and evaders are represented by edges connecting the nodes. An evader is captured if he and one of the pursuers occupy the same node. A question ....

....of the search number s(G) of a given graph G. By the search number it is meant the smallest number of pursuers needed to capture a single evader in finite time, regardless of how the evader decides to move. It turns out that determining if s(G) is smaller than a given constant is NP hard [2, 3]. Pursuit evasion games on graphs have been limited to worst case motions of the evaders. This research was supported by the Office of Naval Research (grant N00014 97 1 0946) When a region in which the pursuit takes place is abstracted to a finite graph, the sensing capabilities of each ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. Journal of the ACM, 35(1):18--44, Jan. 1988.

....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) 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 ....

....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 ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 35:18--44, 1988.

....in this paper is an extension or combination of problems that have been considered in several contexts. Interesting results have been obtained for pursuitevasion in a graph, in which the pursuers and evader can move from vertex to vertex until eventually a pursuer and evader lie in the same vertex [18, 21]. The search number of a graph refers to the minimum number of pursers needed to solve a pursuit evasion problem, and has been closely related to other graph properties such as cutwidth [17, 19] Pursuit evasion scenarios in continuous spaces have arisen in a variety of applications such as air ....

....free spaces is whether there actually exist problems that require a logarithmic number of pursuers. Some results from graph searching will first be described and utilized to construct difficult worst case problem instances. Let Parsons problem refer to the graph searching problem presented in [18, 21]. The task is to specify the number of pursuers required to find an evader that can execute continuous motions along the edges of a graph. Instead of using visibility, capture is achieved when one of the pursuers touches the evader. Let G represent a graph, and S(N) represent the number of ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 35(1):18--44, January 1988.

....this paper is an extension or combination of problems that have been considered in several contexts. Interesting results have been obtained for pursuit evasion in a graph, in which the pursuers and evader can move from vertex to vertex until eventually a pursuer and evader lie in the same vertex [16, 19]. The search number of a graph refers to the minimum number of pursers needed to solve a pursuit evasion problem, and has been closely related to other graph properties such as cutwidth [15, 17] Pursuit evasion scenarios in continuous spaces have arisen in a variety of applications such as air ....

....free spaces is whether there actually exist problems that require a logarithmic number of pursuers. Some results from graph searching will first be described and utilized to construct difficult worst case problem instances. Let Parsons problem refer to the graphsearching problem presented in [16, 19]. The task is to specify the number of pursuers required to find an evader that can execute continuous motions along the edges of a graph. Instead of using visibility, capture is achieved when one of the pursuers touches the evader. Let G represent a graph, and S(G) represent the number of ....

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 35(1):18--44, January 1988.

No context found.

Megiddo, Hakimi, Garey, Johnson, and Papadimitriou. The complexity searching graph. the January 1988.

No context found.

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 1988.

No context found.

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. Journal of the ACM, 35(1):18--44, January 1988.

No context found.

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 1988.

No context found.

N. Megiddo, S. Hakimi, M. Garey, D. Johnson and C. Papadimitriou. The complexity of searching a graph. Journal of the ACM 35(1):18--44, 1988.

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N. Meggido, S. Hakimi, M. Garey, D. Johnson, and C. Papadimitriou. The complexity of searching a graph. JACM 35, 18--44, 1988.

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N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. Assoc. Comput. Mach., 35(1):18--44, 1988.

No context found.

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 35:18--44, 1988.

No context found.

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 1988.

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N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. On the complexity of searching a graph. J. ACM, 35:18-44, 1988.

No context found.

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 1988.

No context found.

N. Megiddo, S. Hakimi, M. Garey, D. Johnson and C. Papadimitriou. The complexity of searching a graph. Journal of the ACM, 35(1):18--44, 1988.

No context found.

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 1988.

No context found.

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 1988.

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N. Megiddo, S.L. Hakimi, M. R. Garey, D.S. Johnson, C.H. Papadimitriou, The complexity of searching a graph, J. Assoc. Comput. Mach. 35 (1988), 18-44.

No context found.

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. Journal of the ACM, 35(1):18--44, 1988.

No context found.

N. Megiddo, S. Hakimi, M. Garey, D. Johnson, and C. Papadimitriou. The complexity of searching a graph. Journal of the ACM, 1988.

No context found.

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson and C. H. Papadimitriou, The complexity of searching a graph, Journal of the ACM, 35 (1988), pp. 18--44.

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