D. M. Thilikos. Algorithms and obstructions for linear-width and related search parameterso. Discrete Applied Mathematics, 105:239--271, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....one. Takahashi, Ueno and Kajitani [35] showed that the mixed search number is equal to the proper path width. In [3] Bienstock and Seymour simpli ed the proof of Lapaugh s result [21] stating that there is a minimal search strategy that does not recontaminate any link (see also [2] Thilikos [36] used graph minors to derive a linear time algorithm that checks whether a network has a search number at most 2. For other results on graph searching, the reader is referred to [7, 8, 12, 30, 32] Contributions to related search problems can be found in [6, 25, 33, 34, 38, 39] and the references ....

D. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics 105, 239-271, 2000.

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D. M. Thilikos. Algorithms and obstructions for linear-width and related search parameterso. Discrete Applied Mathematics, 105:239--271, 2000.

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D. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Appl. Math., 105(1-3):239--271, 2000.

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D. M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics, 105:239--271, 2000.

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D. M. Thilikos. Algorithms and obstructions for linear-width and related search parameterso. Discrete Applied Mathematics, 105:239--271, 2000.

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D. M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics, 105:239--271, 2000.

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Dimitrios M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics, 105:239--271, 2000.

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D. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics 105, 239--271, 2000. 18

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Dimitrios M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics, 105:239-271, 2000.

....The second author was supported by the Training and Mobility of Researchers (TMR) Program, EU contract no ERBFMBICT950198) and Seymour in [2] Linear width can be seen as a linear variant of branch width , in the same way as pathwidth can be seen as a linear variant of treewidth . In [25], it is proved that several variants of problems appearing on graph searching can be reduced to the problem of computing linear width. In a graph searching game a graph represents a system of tunnels where an agile, fast, and invisible fugitive is resorting. We desire to capture this fugitive by ....

....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]

D. M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Technical Report UU-CS-97-35, Department of Computer Science, Utrecht University, Utrecht, 1997.

....Let G be a n vertex graph of treewidth w and A(G) d. Let al.so (U, X) be a tree decomposition of G constructed by the O(w( 2(a)n) algorithm of Bodlaender in [7] From Lemma 2. 2, we know that the pathwidth of G is at most (w l) logn and, as linear width(G) pathwidth(G)q 1 (see, e.g. 46] or [47]) we get that linear width(G) w 1)logn 1. From Lemma 5.2 we have that cutwidth(G) linear width(G ) w 1) logn 1. Notice that a vertex ordering of G with minimum cutwidth corre sponds to an edge ordering of G of minimum linear width. Therefore, it is sufficient to check whether ....

D. M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics, 105:239 271, 2000.

.... 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 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 ....

....en Informatique, Universit e Paris Sud, France. http: www.lri.fr pierre. School of Computer Science, Carleton University, Canada. santoro scs.carleton.ca. Departament de Llenguatges i Sistemes Inform atics, Universitat Polit ecnica de Catalunya, Spain. http: www.lsi.upc.es sedthilk. see [7, 28, 37, 39]) In particular, for any xed k, the class of graphs that can be cleared with up to k searchers is minor closed. Therefore, there is a nite number of obstructions for this class [31] Based on this, it is possible to test in linear time whether an arbitrary graph G satis es s(G) k, for a xed ....

D. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics 105, 239-271, 2000.

....must be cleared again. A search without recontamination is called monotone. The node (edge) search number, ns(G) es(G) is de ned similarly to the mixed search number with the di erence that an edge can be cleared only if A (B) happens. The following is a combination of results in [4] and [24]. Theorem 7. For any graph G the following hold: If G p is the graph occurring from G after subdividing its pendant edges, then ms(G) lw(G p ) We call pendant any edge with an endpoint of degree 1. If G e is the graph occurring from G after subdividing each of its edges, then ....

D. M. Thilikos, Algorithms and obstructions for linear-width and related search parameters, Discrete Applied Mathematics, 105 (2000), pp. 239-271.

.... recognition algorithm for G [RS85a, RS95] Unfortunately, nding the obstruction set of a class G is unsolvable in general [FRS87, FL94, van90] and appears to be a hard structural problem even for simple graph classes, largely due to the rapid explosion in the size of the obstructions [TUK94, Ram95, Thi00]. Following a brute force approach, it is possible to build a computer program enumerating graphs and searching among them for obstructions. A crucial drawback of this method is that there is no general way to bound the search space [van90, FL94] more sophisticated methods are also possible ....

Dimitrios M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics, 105:239-271, 2000. 12

.... recognition algorithm for G [RS85a, RS95] Unfortunately, nding the obstruction set of a class G is unsolvable in general [FRS87, FL94, van90] and appears to be a hard structural problem even for simple graph classes, largely due to the rapid explosion in the size of the obstructions [TUK94, Ram95, Thi00]. Following a brute force approach, it is possible to build a computer program enumerating graphs and searching among them for obstructions. A crucial drawback of this method is that there is no general way to bound the search space [van90, FL94] more sophisticated methods are also possible ....

Dimitrios M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics, 105:239-271, 2000.

....n vertex graph of treewidth w and (G) d. Let al.so (U; X) be a tree decomposition of G constructed by the O(w O(w) 2 O(w 3 ) n) algorithm of Bodlaender in [7] From Lemma 2. 2, we know that the pathwidth of G is at most (w 1) log n and, as linear width(G) pathwidth(G) 1 (see, e.g. 44] or [45]) we get that linear width(G) w 1) log n 1. From Lemma 5.2 we have that cutwidth(G) linear width(G D ) w 1) log n 1. Notice that a vertex ordering of G D with minimum cutwidth corresponds to an edge ordering of G n of minimum linear width. Therefore, it is sucient to check ....

D. M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics, 105:239-271, 2000.

....decompositions. Much research has been done towards the construction of linear time algorithms solving Pi d k (T ) and Pi c k (T ) In [5] a linear (on the size of the input) time algorithm for treewidth was constructed. For further results concerning on related graph theoretic parameters see [3, 7, 8, 13, 16, 14, 15, 19, 21, 19, 22, 23, 24]. In this paper, we find analogous results to those of [5] for the parameter of branchwidth. Namely, we prove that, for any fixed k, one can construct a linear 2 time algorithm that solves Pi d k (B) and Pi c k (B) An immediate consequence of this result is that, for any fixed k, one can ....

D. M. Thilikos. Algorithms and obstructions for linear-width and related search parameterso. Discrete Applied Mathematics, 105:239--271, 2000.

....r g. Linear width was first mentioned by Thomas in [30] and is strongly connected with the notion of crusades introduced by Bienstock and Seymour in [2] Linear width can be seen as a linear variant of branch width , in the same way as pathwidth can be seen as a linear variant of treewidth . In [29], it is proved that several variants of problems appearing on graph searching can be reduced to the problem of computing linear width. In a graph searching game a graph represents a system of tunnels where an agile, fast, and invisible fugitive is resorting. We desire to capture this fugitive by ....

....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]

D. M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Technical Report UU-CS-97-35, Department of Computer Science, Utrecht University, Utrecht, 1997.

....y The second author was supported by the Training and Mobility of Researchers (TMR) Program, EU contract no ERBFMBICT950198) and Seymour in [2] Linear width can be seen as a linear variant of branch width , in the same way as pathwidth can be seen as a linear variant of treewidth . In [25], it is proved that several variants of problems appearing on graph searching can be reduced to the problem of computing linear width. In a graph searching game a graph represents a system of tunnels where an agile, fast, and invisible fugitive is resorting. We desire to capture this fugitive by ....

....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]

D. M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Technical Report UU-CS-97-35, Department of Computer Science, Utrecht University, Utrecht, 1997.

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