H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, ICALP'97, pages 627--637. Springer-Verlag, Lecture Notes in Computer Science, Vol. 1256, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....not provide a way to construct such an algorithm but it merely proves existence of one. From the work of Bodlander and Thilikos, we know that an effective decidable algorithm to determine membership in the class of graphs with branchwidth at most k can be constructed and solved in linear time [1]. This algorithm is highly exponential in a polynomial in k and thus is quite impractical for actual implementation and use. Being as such, other ways are needed to characterize graphs with branchwidth at most k. Although there are several different ways to characterize a graph class, we have ....

....two [13] ffl at most 2 if and only if G has no K 4 minor [13] Bodlander made an addition to the list when he determined the obstruction set of graphs with branchwidth at most three. He proved that a graph G has branchwidth at most ffl 3 if and only if G has no K 5 , Q 3 , M 6 , or M 8 minor [1]. K Q M M 5 3 6 8 Figure 3.1: O fi 3 Let G = V (G) E(G) be a graph and S = v 1 ; v 2 ; v 3 ; v 4 ) be a subset of V (G) The set S is a cross if S i = S Gamma v i , 1 i 4 , are all minimal separators of G. Bodlaender used the fact that graphs with branchwidth at most three have graphs ....

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H. L. Bodlaender and D. M. Thilkos. Constructive linear time algorithms for branchwidth. Lecture Notes in Computer Science, 1997.

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H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, ICALP'97, pages 627--637. Springer-Verlag, Lecture Notes in Computer Science, Vol. 1256, 1997.

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Bodlaender, H. L., and Thilikos, D. M. Constructive linear time algorithms for branchwidth. In Proceedings 24th International Colloquium on Automata, Languages, and Programming (1997), P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, Eds., Springer Verlag, Lecture Notes in Computer Science, vol. 1256, pp. 627--637.

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D. M. Thilikos and H. L. Bodlaender. Constructive linear time algorithms for branchwidth. Technical Report UU-CS-2000-38, Dept. of Computer Science, Utrecht University, 2000.

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H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In Automata, languages and programming (Bologna, 1997), volume 1256 of Lecture Notes in Computer Science, pages 627--637. Springer, Berlin, 1997.

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H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, ICALP'97, pages 627--637. Springer-Verlag, Lecture Notes in Computer Science, Vol. 1256, 1997.

....and then apply an algorithm, presented in this paper, that, given such a path decomposition, solves our problem. It can actually be avoided to work with tree decompositions by a modification of the algorithm in [3] An other parameter related to linear width is branch width. In another paper [5], we give a similar algorithm for branch width. That algorithm uses the techniques of this paper as a building block for a more complicated algorithm. This paper is organised as follows. In Section 2, we give some preliminary results and definitions. The main algorithm is presented in Section 3. ....

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, pages 627--637. Springer Verlag, Lecture Notes in Computer Science, vol. 1256, 1997.

.... given a graph parameter f that is closed under taking of minors, each value of k corresponds to a different obstruction set, i.e. ob(G[f; k] To our knowledge, obstruction sets have been found for the following graph parameters: treewidth, for k 3 (see [1, 18, 32] branchwidth, for k 3 (see [8]) node search number, for k 3 (see [10, 20] and mixed search number, for k 2 (see [34] The linear width of a graph G is defined to be the least integer k such that the edges of G can be arranged in a linear ordering (e 1 ; e r ) in such a way that for every i = 1; r Gamma ....

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, pages 627--637. Springer Verlag, Lecture Notes in Computer Science, vol. 1256, 1997.

....19 and Theorem 10. 26 5 Conclusions The techniques described here only deal with problems where a linear order has to be found, and as common characteristic we have that yes instances have bounded pathwidth. Similar algorithms are known however for graphs of bounded treewidth (like branchwidth [8], carving width [20] and treewidth itself [15, 7] In order to be able to present or extend such algorithms like we did above, additional techniques have to be added to the machinery. In particular, there are two additional complications that must be mastered: The desired output of the problem ....

Bodlaender, H. L., and Thilikos, D. M. Constructive linear time algorithms for branchwidth. In Proceedings 24th International Colloquium on Automata, Languages, and Programming (1997), P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, Eds., Springer Verlag, Lecture Notes in Computer Science, vol. 1256, pp. 627-637.

....k constant. Finally, in [22] an explicit constructive linear time algorithm was presented able to output the optimal vertex ordering in case of a positive answer. This algorithm is based on the fact that graphs with small cutwidth have also small pathwidth and develops further the techniques in [12, 13, 14] in order to use a bounded width path decomposition for computing the cutwidth of G. This paper extends the algorithm in [22] in the sense that it uses all of its subroutines and it solves the problem of [22] for graphs with bounded treewidth. Although this extension is not really useful for the ....

....of its correctness. Subsections 2.1 and 2.2 contain the definitions of treewidth, pathwidth and cutwidth. Most of the preliminary results of Subsection 2. 3, concern operations on sequences of integers and the definitions of the most elementary of them was introduced in [22] and [12] see also [13, 14]) Also, the main tool for exploiting the small treewidth of the input graph is the notion of the characteristic of a vertex ordering, introduced in [22] and defined in Subsection 2.5 of this paper. For the above reasons, we use notation compatible with the one used in [22] Algorithm Join Node ....

[Article contains additional citation context not shown here]

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. Technical Report UU-CS-2000-38, Dept. of Computer Science, Utrecht University, 2000.

....19 and Theorem 10. 21 5 Conclusions The techniques described here only deal with problems where a linear order has to be found, and as common characteristic we have that yes instances have bounded pathwidth. Similar algorithms are known however for graphs of bounded treewidth (like branchwidth [8], carving width [20] and treewidth itself [15, 7] In order to be able to present or extend such algorithms like we did above, additional techniques have to be added to the machinery. In particular, there are two additional complications that must be mastered: The desired output of the problem ....

Bodlaender, H. L., and Thilikos, D. M. Constructive linear time algorithms for branchwidth. In Proceedings 24th International Colloquium on Automata, Languages, and Programming (1997), P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, Eds., Springer Verlag, Lecture Notes in Computer Science, vol. 1256, pp. 627-637.

....first try to find a tree decomposition of small width, and then utilise the advantages of the tree structure of the decomposition. This paper is the full version of part of the paper titled Constructive Linear Time Algorithms for Branchwidth which appeared in the proceedings of ICALP 97 (see [7]) y This research was partially supported by ESPRIT Long Term Research Project 20244 (project ALCOM IT: Algorithms and Complexity in Information Technology) The second author was supported by the Training and Mobility of Researchers (TMR) Program, EU contract no ERBFMBICT950198) The ....

.... tailor made algorithms have been presented for small values of k: treewidth 1 and 2 [12, 20] treewidth 3 [3, 10, 12] treewidth 4 [16] Also, the obstruction sets for treewidth 1, 2, and 3 are known [4, 17, 20] Recently, a linear time algorithm solving Pi k (B) and Pi k (B) was given in [7]. Unfortunately, the algorithms in [7] appear (similarly to the case of treewidth) to be non practical. In this paper, we provide special tailor made results for the case where k 3. More specifically, for the class of graphs with branchwidth 3, we identify the obstruction set and we give a ....

[Article contains additional citation context not shown here]

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, pages 627--637. Springer Verlag, Lecture Notes in Computer Science, vol. 1256, 1997.

....k constant. Finally, in [47] an explicit constructive linear time algorithm was presented able to output the optimal vertex ordering in case of a positive answer. This algorithm is based on the fact that graphs with small cutwidth have also small pathwidth and develops further the techniques in [12, 13, 46] in order to use a bounded width path decomposition for computing the cutwidth of G. This paper extends the algorithm in [47] in the sense that it uses all of its subroutines and it solves the problem of [47] for graphs with bounded treewidth. Although this extension is not really useful for the ....

....proof of its correctness. Subsections 2.1 and 2.2 contain the de nitions of treewidth, pathwidth and cutwidth. Most of the preliminary results of Subsection 2. 3, concern operations on sequences of integers and the de nitions of the most elementary of them was introduced in [47] and [12] see also [13, 46]) Also, the main tool for exploiting the small treewidth of the input graph is the notion of the characteristic of a vertex ordering, introduced in [47] and de ned in Subsection 2.5 of this paper. For the above reasons, we use notation compatible with the one used in [47] Algorithm Join Node ....

[Article contains additional citation context not shown here]

D. M. Thilikos and H. L. Bodlaender. Constructive linear time algorithms for branchwidth. Technical Report UU-CS-2000-38, Dept. of Computer Science, Utrecht University, 2000.

....immersion obstruction set for the class of the graphs with cutwidth (carving width) at most k. We mention that optimal constructive results exist so far only for minor closed parameters such as treewidth and pathwidth [2, 4] agile graph searching parameters [6] linear width [6] and branchwidth [5]. Besides the fact that that our techniques are motivated by those used for the aforementioned minor closed parameters, in our knowledge, our results are the rst concerning immersion closed parameters and we believe that our approach is applicable to other parameters as well (e.g. Modified ....

....algorithms In this section, we will describe the general structure of both of our algorithms along with their basic mathematical concepts. A key tool in both cases already used in the bibliography for other parameters like pathwidth and treewidth in [2] linear width in [6] and branchwidth in [5], is the notion of characteristics. In a few words, a characteristic serves as a mathematical tool that lters the data of the main structure of a parameter to its essential part, that is, the part able to reproduce it with respect to a node i of the path (tree) decomposition. Moreover, as we will ....

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, ICALP'97, pages 627-637. Springer-Verlag, Lecture Notes in Computer Science, Vol. 1256, 1997.

....immersion obstruction set for the class of the graphs with cutwidth (carving width) at most k. We mention that optimal constructive results exist so far only for minor closed parameters such as treewidth and pathwidth [2, 4] agile graph searching parameters [6] linear width [6] and branch width [5]. Besides the fact that that our techniques are motivated by those used for the aforementioned minor closed parameters, in our knowledge, our results are the first concerning immersion closed parameters and we believe that our approach is applicable to other parameters as well (e.g. Modified ....

....algorithms In this section, we will describe the general structure of both of our algorithms along with their basic mathematical concepts. A key tool in both cases already used in the bibliography for other parameters like pathwidth and treewidth in [2] linear width in [6] and branchwidth in [5], is the notion of characteristics. In a few words, a characteristic serves as a mathematical tool that filters the data of the main structure of a parameter to its essential part, that is, the part able to reproduce it with respect to a node i of the path (tree) decomposition. Moreover, as we ....

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, ICALP'97, pages 627--637. Springer-Verlag, Lecture Notes in Computer Science, Vol. 1256, 1997.

....of a linear time algorithm that checks whether an input graph G has cutwidth k and, if this is the case, it further outputs a vertex layout of G of minimum cutwidth. The key tool in our algorithm, as for other parameters like pathwidth and treewidth in [4] linear width in [6] and branchwidth in [5], is the notion of characteristics. In a few words, a characteristic serves to filter the main data structure of a parameter to its essential part, a part that is able to be constructed from node to node of a path decomposition. Moreover, as we will see, the information encoded by a characteristic ....

....is able to determine the immersion obstruction set for the class of the graphs with cutwidth at most k. We mention that optimal constructive results exist so far only for minor closed parameters such as treewidth and pathwidth [4, 3] agile search parameters [6] linearwidth [6] and branch width [5]. Besides the fact that our techniques are motivated by those used in the aforementioned minor closed parameters, in our knowledge, our results are the first concerning immersion closed parameters and we believe that our approach is applicable to other parameters as well (e.g. Modified Cutwidth, ....

[Article contains additional citation context not shown here]

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, ICALP'97, pages 627--637. Springer-Verlag, Lecture Notes in Computer Science, Vol. 1256, 1997.

....immersion obstruction set for the class of the graphs with cutwidth (carving width) at most k. We mention that optimal constructive results exist so far only for minor closed parameters such as treewidth and pathwidth [2, 4] agile graph searching parameters [6] linear width [6] and branchwidth [5]. Besides the fact that that our techniques are motivated by those used for the aforementioned minor closed parameters, in our knowledge, our results are the first concerning immersion closed parameters and we believe that our approach is applicable to other parameters as well (e.g. Modified ....

....algorithms In this section, we will describe the general structure of both of our algorithms along with their basic mathematical concepts. A key tool in both cases already used in the bibliography for other parameters like pathwidth and treewidth in [2] linear width in [6] and branchwidth in [5], is the notion of characteristics. In a few words, a characteristic serves as a mathematical tool that filters the data of the main structure of a parameter to its essential part, that is, the part able to reproduce it with respect to a node i of the path (tree) decomposition. Moreover, as we ....

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, ICALP'97, pages 627--637. Springer-Verlag, Lecture Notes in Computer Science, Vol. 1256, 1997.

....of a linear time algorithm that checks whether an input graph G has cutwidth# k and, if this is the case, it further outputs a vertex layout of G of minimum cutwidth. The key tool in our algorithm, as for other parameters like pathwidth and treewidth in [4] linear width in [6] and branchwidth in [5], is the notion of characteristics. In a few words, a characteristic serves to filter the main data structure of a parameter to its essential part, a part that is able to be constructed from node to node of a path decomposition. Moreover, as we will see, the information encoded by a characteristic ....

....is able to determine the immersion obstruction set for the class of the graphs with cutwidth at most k. We mention that optimal constructive results exist so far only for minor closed parameters such as treewidth and pathwidth [4, 3] agile search parameters [6] linearwidth [6] and branch width [5]. Besides the fact that our techniques are motivated by those used in the aforementioned minor closed parameters, in our knowledge, our results are the first concerning immersion closed parameters and we believe that our approach is applicable to other parameters as well (e.g. Modified Cutwidth, ....

[Article contains additional citation context not shown here]

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, ICALP'97, pages 627--637. Springer-Verlag, Lecture Notes in Computer Science, Vol. 1256, 1997.

....and then apply an algorithm, presented in this paper, that, given such a path decomposition, solves our problem. It can actually be avoided to work with tree decompositions by a modification of the algorithm in [3] An other parameter related to linear width is branch width. In another paper [6], we give a similar algorithm for branch width. That algorithm uses the techniques of this paper as a building block for a more complicated algorithm. This paper is organised as follows. In Section 2, we give some preliminary results and definitions. The main algorithm is presented in Section 3. ....

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, pages 627--637. Springer Verlag, Lecture Notes in Computer Science, vol. 1256, 1997.

.... given a graph parameter f that is closed under taking of minors, each value of k corresponds to a different obstruction set, i.e. ob(G[f; k] To our knowledge, obstruction sets have been found for the following graph parameters: treewidth, for k 3 (see [1, 18, 32] branchwidth, for k 3 (see [8]) node search number, for k 3 (see [10, 20] and mixed search number, for k 2 (see [34] The linear width of a graph G is defined to be the least integer k such that the edges of G can be arranged in a linear ordering (e 1 ; e r ) in such a way that for every i = 1; r Gamma ....

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, pages 627--637. Springer Verlag, Lecture Notes in Computer Science, vol. 1256, 1997.

....and then apply an algorithm, presented in this paper, that, given such a path decomposition, solves our problem. It can actually be avoided to work with tree decompositions by a modification of the algorithm in [3] An other parameter related to linear width is branch width. In another paper [5], we give a similar algorithm for branch width. That algorithm uses the techniques of this paper as a building block for a more complicated algorithm. This paper is organised as follows. In Section 2, we give some preliminary results and definitions. The main algorithm is presented in Section 3. ....

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, pages 627--637. Springer Verlag, Lecture Notes in Computer Science, vol. 1256, 1997.

....first try to find a tree decomposition of small width, and then utilise the advantages of the tree structure of the decomposition. This paper is the full version of part of the paper titled Constructive Linear Time Algorithms for Branchwidth which appeared in the proceedings of ICALP 97 (see [7]) y This research was partially supported by ESPRIT Long Term Research Project 20244 (project ALCOM IT: Algorithms and Complexity in Information Technology) z The second author was supported by the Training and Mobility of Researchers (TMR) Program, EU contract no ERBFMBICT950198) The ....

.... algorithms have been presented for small values of k: treewidth 1 and 2 [12, 20] treewidth 3 [3, 10, 12] treewidth 4 [16] Also, the obstruction sets for treewidth 1, 2, and 3 are known [4, 17, 20] Recently, a linear time algorithm solving Pi d k (B) and Pi c k (B) was given in [7]. Unfortunately, the algorithms in [7] appear (similarly to the case of treewidth) to be non practical. In this paper, we provide special tailor made results for the case where k 3. More specifically, for the class of graphs with branchwidth 3, we identify the obstruction set and we give a ....

[Article contains additional citation context not shown here]

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, pages 627--637. Springer Verlag, Lecture Notes in Computer Science, vol. 1256, 1997.

....first try to find a tree decomposition of small width, and then utilise the advantages of the tree structure of the decomposition. This paper is the full version of part of the paper titled Constructive Linear Time Algorithms for Branchwidth which appeared in the proceedings of ICALP 97 (see [7]) y This research was partially supported by ESPRIT Long Term Research Project 20244 (project ALCOM IT: Algorithms and Complexity in Information Technology) z Department of Computer Science, Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, the Netherlands, e mail: hansb cs.uu.nl x ....

.... for small values of k: treewidth 1 and 2 [14, 21] treewidth 3 [3, 12, 14] treewidth 4 [17] Also, the obstruction sets for the class of graphs with treewidth 1, 2, and 3 are known [4, 18, 21] Recently, a linear time algorithm solving Branchwidth and its constructive version was given in [7]. Unfortunately, the algorithms in [7] appear to be impractical, similarly to the case of treewidth. In this paper, we provide special tailor made results for the cases where k 3. As the cases where k 2 are trivial we focus our attention to the case k = 3. More specifically, for the class of ....

[Article contains additional citation context not shown here]

H. L. Bodlaender and D. M. Thilikos. Constructive linear time algorithms for branchwidth. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings 24th International Colloquium on Automata, Languages, and Programming, pages 627--637. Springer Verlag, Lecture Notes in Computer Science, vol. 1256, 1997.

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