Abstract
Given an arbitrary graph G and a number k, it is well-known by a result of Seymour and Thomas [22] that G has treewidth strictly larger than k if and only if it has a bramble of order k + 2. Brambles are used in combinatorics as certificates proving that the treewidth of a graph is large. From an algorithmic point of view there are several algorithms computing tree-decompositions of G of width at most k, if such decompositions exist and the running time is polynomial for constant k. Nevertheless, when the treewidth of the input graph is larger than k, to our knowledge there is no algorithm constructing a bramble of order k + 2. We give here such an algorithm, running in \({\mathcal O}(n^{k+4})\) time. For classes of graphs with polynomial number of minimal separators, we define a notion of compact brambles and show how to compute compact brambles of order k + 2 in polynomial time, not depending on k.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Amini, O., Mazoit, F., Thomassé, S., Nisse, N.: Partition Submodular Functions. To appear in Disc. Math. (2008), http://www.lirmm.fr/~thomasse/liste/partsub.pdf
Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of Finding Embeddings in a k-Tree. SIAM J. on Algebraic and Discrete Methods 8, 277–284 (1987)
Bachoore, E.H., Bodlaender, H.L.: New Upper Bound Heuristics for Treewidth. In: Nikoletseas, S.E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 216–227. Springer, Heidelberg (2005)
Bellenbaum, P., Diestel, R.: Two Short Proofs Concerning Tree-Decompositions. Combinatorics, Probability & Computing 11(6) (2002)
Bodlaender, H.L.: A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth. SIAM J. Comput. 25, 1305–1317 (1996)
Bodlaender, H.L.: A Partial k-Arboretum of Graphs with Bounded Treewidth. Theor. Comput. Sci. 209(1-2), 1–45 (1998)
Bodlaender, H.L., Grigoriev, A., Koster, A.M.C.A.: Treewidth Lower Bounds with Brambles. Algorithmica 51(1), 81–98 (2008)
Bodlaender, H.L., Koster, A.M.C.A.: On the Maximum Cardinality Search Lower Bound for Treewidth. Disc. App. Math. 155(11), 1348–1372 (2007)
Bodlaender, H.L., Wolle, T., Koster, A.M.C.A.: Contraction and Treewidth Lower Bounds. J. Graph Algorithms Appl. 10(1), 5–49 (2006)
Bouchitté, V., Todinca, I.: Treewidth and Minimum Fill-in: Grouping the Minimal Separators. SIAM J. Comput. 31(1), 212–232 (2001)
Bouchitté, V., Todinca, I.: Listing All Potential Maximal Cliques of a Graph. Theor. Comput. Sci. 276(1-2), 17–32 (2002)
Chapelle, M., Mazoit, F., Todinca, I.: Constructing Brambles. Technical Report, LIFO, Université d’Orléans (2009)
Clautiaux, F., Carlier, J., Moukrim, A., Nègre, S.: New Lower and Upper Bounds for Graph Treewidth. In: Jansen, K., Margraf, M., Mastrolli, M., Rolim, J.D.P. (eds.) WEA 2003. LNCS, vol. 2647, pp. 70–80. Springer, Heidelberg (2003)
Fomin, F.V., Fraigniaud, P., Nisse, N.: Nondeterministic Graph Searching: From Pathwidth to Treewidth. Algorithmica 53(3), 358–373 (2009)
Fomin, F.V., Kratsch, D., Todinca, I., Villanger, Y.: Exact Algorithms for Treewidth and Minimum Fill-in. SIAM J. Comput. 38(3), 1058–1079 (2008)
Fomin, F.V., Thilikos, D.M.: An Annotated Bibliography on Guaranteed Graph Searching. Theor. Comput. Sci. 399(3), 236–245 (2008)
Grohe, M., Marx, D.: On Tree Width, Bramble Size, and Expansion. J. Comb. Theory, Ser. B 99(1), 218–228 (2009)
Lyaudet, L., Mazoit, F., Thomassé, S.: Partitions Versus Sets: A Case of Duality (submitted 2009), http://www.lirmm.fr/~thomasse/liste/dualite.pdf
Mazoit, F.: Décompositions Algorithmiques des Graphes. PhD thesis, École Normale Supérieure de Lyon (2004) (in French)
Mazoit, F.: The Branch-width of Circular-Arc Graphs. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 727–736. Springer, Heidelberg (2006)
Robertson, N., Seymour, P.D.: Graph Minors X. Obstructions to Tree Decompositions. J. Comb. Theory, Ser. B 52, 153–190 (1991)
Seymour, P.D., Thomas, R.: Graph Searching and a Min-Max Theorem for Tree-Width. J. Comb. Theory, Ser. B 58(1), 22–33 (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chapelle, M., Mazoit, F., Todinca, I. (2009). Constructing Brambles. In: Královič, R., Niwiński, D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03816-7_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-03816-7_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03815-0
Online ISBN: 978-3-642-03816-7
eBook Packages: Computer ScienceComputer Science (R0)