Abstract
In this paper we show that the treewidth and pathwidth of a permutation graph can be computed in polynomial time. In fact we show that, for permutation graphs, the treewidth and pathwidth are equal. These results make permutation graphs one of the few non-trivial graph classes for which at the moment, treewidth is known to be computable in polynomial time. Our algorithm to decide whether the treewidth (pathwidth) is at most some given integer k, can be implemented to run in O(nk 2) time, when the matching diagram is given. We show that this algorithm can easily be adapted to compute the pathwidth of a permutation graph in O(nk 2) time, where k is the pathwidth.
This author is supported by the foundation for Computer Science (S.I.O.N) of the Netherlands Organization for Scientific Research (N.W.O.).
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
S. Arnborg, Efficient algorithms for combinatorial problems on graphs with bounded decomposability-A survey. BIT 25, 2–23, 1985.
S. Arnborg, D.G. Cornell and A. Proskurowski, Complexity of finding embeddings in a k-tree, SIAM J. Alg. Disc. Meth. 8, 277–284, 1987.
S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees. Disc. Appl. Math. 23, 11–24, 1989.
C. Berge and C. Chvatal, Topics on Perfect Graphs, Annals of Discrete Math. 21, 1984.
H.L. Bodlaender, A tourist guide through treewidth, Technical report RUU-CS-92-12, Department of Computer Science, Utrecht University, Utrecht, The Netherlands, 1992. To appear in: Proceedings 7th Int. Meeting of Young Computer Scientists.
H. Bodlaender and R.H. Möhring, The pathwidth and treewidth of cographs, In Proceedings 2nd Scandinavian Workshop on Algorithm Theory, 301–309, Springer Verlag, Lect. Notes in Comp. Sc., vol. 447, 1990. To appear in: SIAM J. Discr. Math.
H. Bodlaender and T. Kloks, Better algorithms for the pathwidth and treewidth of graphs, Proceedings of the 18th Int. Coll. on Automata, Languages and Programming, 544–555, Springer Verlag, Lect. Notes in Comp. Sc., vol. 510, 1991.
A. Brandstädt, Special graph classes — a survey, Schriftenreihe des Fachbereichs Mathematik, SM-DU-199 (1991) Universität Duisburg Gesamthochschule.
S. Even, A. Pnueli and A. Lempel, Permutation graphs and transitive graphs, J. Assoc. Comput. Mach. 19, 400–410, 1972.
M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
J. Gustedt, Pathwidth for chordal graphs is NP-complete. To appear in: Discr. Appl. Math.
T. Kloks and H. Bodlaender, Approximating treewidth and pathwidth of some classes of perfect graphs. 3th Ann. Int. Symp. on Algorithms and Computation (ISAAC'92), 116–125, Springer Verlag, Lect. Notes in Comp. Sc., vol. 650, 1993.
T. Kloks and D. Kratsch, Treewidth of chordal bipartite graphs. In: Proc. 10th Ann. Symp. on Theoretical Aspects of Computer Science, 80–89, Springer Verlag, Lect. Notes in Comp. Sc., vol. 665, 1993.
J. Lagergren and S. Arnborg, Finding minimal forbidden minors using a finite congruence, Proceedings of the 18th Int. Coll. on Automata, Languages and Programming, 532–543, Springer Verlag, Lect. Notes in Comp. Sc., vol. 510, 1991.
J. van Leeuwen, Graph algorithms. In Handbook of Theoretical Computer Science, A: Algorithms an Complexity Theory, 527–631, Amsterdam, 1990. North Holland Publ. Comp.
C.G. Lekkerkerker and J.Ch. Boland, Representation of a finite graph by a set of intervals on the real line, Fund. Math. 51, 45–64, 1962.
B. Reed, Finding approximate separators and computing treewidth quickly, Proc. 24th Ann. ACM Symp. on Theory of Computing, 221–228, 1992.
J. Spinrad, On comparability and permutation graphs, SIAM J. Comp. 14, 658–670, 1985.
R. Sundaram, K. Sher Singh and C. Pandu Rangan, Treewidth of circular arc graphs, Manuscript, 1991, to appear in SIAM J. Disc. Math.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bodlaender, H., Kloks, T., Kratsch, D. (1993). Treewidth and pathwidth of permutation graphs. In: Lingas, A., Karlsson, R., Carlsson, S. (eds) Automata, Languages and Programming. ICALP 1993. Lecture Notes in Computer Science, vol 700. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56939-1_66
Download citation
DOI: https://doi.org/10.1007/3-540-56939-1_66
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56939-8
Online ISBN: 978-3-540-47826-3
eBook Packages: Springer Book Archive