Abstract
We present a new lower bound for the treewidth (and hence the pathwidth) of a graph and give a linear-time algorithm to compute the bound. With the growing interest in treewidth based methods, this bound has many potential applications.
Our bound helps shed new light on the structure of obstructions for width ω. As a result, we are able to characterize completely those treewidth obstructions of order ω+3. Unexpectedly, we find that these graphs are exactly the pathwidth obstructions of order ω+3. Further, we are also able to enumerate these obstructions.
Surprisingly, while there is only one obstruction of order ω+2 for width ω, we find that the number of obstructions of order ω+3 alone is an asymptotically exponential function of ω. Our proof of this is based on the theory of partitions of integers and is the first non-trivial lower bound on the number of obstructions for treewidth.
Preview
Unable to display preview. Download preview PDF.
References
S. Arnborg, D. Corneil, and A. Proskurowski, “Complexity of finding embeddings in a k-tree,” SIAM J. Alg. Disc. Meth. 8 (1987), 277–284.
S. Arnborg, J. Lagergren, and D. Seese, “Problems easy for tree-decomposable graphs,” Journal of Algorithms 12 (1991), 308–340.
G. E. Andrews, “The Theory of Partitions,” in Gian-Carlo Rota, Editor, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley, 1976.
S. Arnborg and A. Proskurowski, “Linear time algorithms for NP-hard problems restricted to partial k-trees,” Discrete Applied Math. 23 (1989), 11–24.
S. Arnborg, A. Proskurowski, and D. Corneil, “Forbidden minors characterization of partial 3-Trees,” Discrete Mathematics 80 (1990), 1–19.
H. L. Bodlaender, “A linear time algorithm for finding tree-decompositions of small treewidth,” Proceedings, 25th ACM Symposium on Theory of Computing (1993), 226–234.
H. L. Bodlaender, “A tourist guide through treewidth,” Acta Cybernetica 11 (1993), 1–23.
H. L. Bodlaender and T. Kloks, “Better algorithms for the pathwidth and treewidth of graphs,” Proceedings, 18th ICALP, Lecture Notes in Computer Science 510 (1991), 544–555.
W. W-M. Dai and M. Sato, “Minimal forbidden minor characterization of planar 3-trees and application to circuit layout,” Proceedings, IEEE International Symposium on Circuits and Systems (1990), 2677–2681.
M. R. Fellows and M. A. Langston, “Nonconstructive tools for proving polynomial time decidability,” Journal of the ACM, 35:3 (1988), 727–739.
M. R. Fellows and M. A. Langston, “An analogue of the Myhill-Nerode theorem and its use in computing finite-basis characterizations,” Proceedings, 30th Symposium on Foundations of Computer Science (1989), 520–525.
R. Govindan, M. Langston, and S. Ramachandramurthi, “A practical approach to layout optimization,” Proceedings, 6th International Conference on VLSI Design (1993), 222–225.
N. G. Kinnersley and M. A. Langston, “Obstruction set isolation for the Gate Matrix Layout problem,” Annals of Discrete Mathematics, to appear.
A. Kornai and Z. Tuza, “Narrowness, pathwidth, and their application in natural language processing,” Discrete Applied Mathematics 36 (1992), 87–92.
J. Lagergren, “An upper bound on the size of an obstruction,” in Graph Structure Theory, N. Robertson and P. Seymour (editors), Contemporary Mathematics 147 (1993), 601–621.
J. Lagergren and S. Arnborg, “Finding minimal forbidden minors using a finite congruence,” Proceedings, 18th ICALP, Lecture Notes in Computer Science 510 (1991), 533–543.
B. Reed, “Finding approximate separators and computing treewidth quickly,” Proceedings, 24th ACM Symposium on Theory of Computing (1992), 221–228.
N. Robertson and P. D. Seymour, “Graph Minors II. Algorithmic aspects of treewidth,” Journal of Algorithms 7 (1986), 309–322.
N. Robertson and P. D. Seymour, “Graph Minors IV. Tree-Width and Well-Quasi-Ordering,” J. of Combinatorial Theory, Series B 48 (1990), 227–254.
N. Robertson and P. D. Seymour, “Graph Minors XIII. The Disjoint Paths Problem,” manuscript (1986).
D. P. Sanders, “On linear recognition of treewidth at most four,” manuscript (1992).
A. Satyanarayana and L. Tung, “A characterization of partial 3-trees,” Networks 20 (1990), 299–322.
X. Yan, “A relative approximation algorithm for computing the pathwidth,” Master's Thesis, Department of Computer Science, Washington State University, Pullman (1989).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ramachandramurthi, S. (1995). A lower bound for treewidth and its consequences. In: Mayr, E.W., Schmidt, G., Tinhofer, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 1994. Lecture Notes in Computer Science, vol 903. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59071-4_34
Download citation
DOI: https://doi.org/10.1007/3-540-59071-4_34
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59071-2
Online ISBN: 978-3-540-49183-5
eBook Packages: Springer Book Archive