Abstract
We show that all minimal a-b separators (vertex sets) disconnecting a pair of given non-adjacent vertices a and b in an undirected and connected graph with n vertices can be computed in O(n 2 R ab ) time, where R ab is the number of minimal a-b separators. This result matches the known worst-case time complexity of its counterpart problem of computing all a-b cutsets (edge sets) [13] and solves an open problem posed in [11].
Author for correspondence. This work was done when the author was visiting Hong Kong Baptist University in 1998 and was partially supported by the Australia Research Council under its Small Grants Scheme.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
H. Ariyoshi, Cut-set graph and systematic generation of separating sets, IEEE Trans. Circuit Theory CT-19(3) (1972) 233–240.
S. Arnberg, Efficient algorithms for combinatorial problems on graphs with bounded decomposability-A survey, BIT 25 (1985) 2–23.
A. Gibbons, Algorithmic Graph Theory, Cambridge Univ. Press, 1985.
L.A. Goldberg, Efficient Algorithms for Listing Combinatorial Structures, Cambridge Univ. Press, 1993.
M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
D.S. Hirschberg, A.K. Chandra and D.V. Sarwate, Computing connected components on parallel computers, Commu. ACM 22(8) (1979) 461–464.
A. Kanevsky, On the number of minimum size separating vertex sets in a graph and how to find all of them, Proc. 1st Ann. ACM-SIAM Symp. Discrete Algorithms (1990) 411–421.
T. Kloks and D. Kratsh, Finding all minimal separators of a graph, Proc. Theoretical Aspects of Computer Sci. LNCS 775 (1994) 759–767.
D.E. Knuth, The Art of Computer Programming, Vol 3: Sorting and Searching, Addison-Wesley, 1973.
E.M. Reingold, J. Nievergelt and N. Deo, Combinatorial Algorithms: Theory and Practice, Prentice-Hall, Englewood Cliffs (1977).
H. Shen and W. Liang, Efficient enumeration of all minimal separators in a graph, Theoretical Computer Science 180 (1997) 169–180.
H. Shen, K. Li and S.Q. Zheng, Separators are as simple as cutsets (long version), manuscript.
S. Tsukiyama, I. Shirakawa and H. Ozaki, An algorithm to enumerate all cutsets of a graph in linear time per cutest, J. ACM 27(4) (1980) 619–632.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Shen, H., Li, K., Zheng, SQ. (1999). Separators Are as Simple as Cutsets. In: Thiagarajan, P.S., Yap, R. (eds) Advances in Computing Science — ASIAN’99. ASIAN 1999. Lecture Notes in Computer Science, vol 1742. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46674-6_29
Download citation
DOI: https://doi.org/10.1007/3-540-46674-6_29
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66856-5
Online ISBN: 978-3-540-46674-1
eBook Packages: Springer Book Archive