Abstract
We prove that every n-vertex graph of genus g and maximal degree k has an edge separator of size O(√gkn). The upper bound is best possible to within a constant factor. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separator to the isoperimetric number problem, graph embeddings and lower bounds for crossing numbers.
Both authors were supported by a research grant from Humboldt Foundation, Bonn, Germany
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Boshier, D. O., Enlarging properties of graphs, Ph.D. Thesis, Royal Holloway and Bedford New Colledge, University of London, 1987.
Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian, In: Problems in Analysis, Princeton University Press, 1970, 159–199.
De Millo, R.A., Eisenstat, S. C., Lipton, R. J., Preserving average proximity in arrays, Communications of the ACM, 21, 1978, 228–230.
Diks, K., Djidjev, H. N., Sýkora, O., Vrťo, I., Edge separators for planar graphs and their applications, In: Proc. of the 13-th Symposium on Mathematical Foundations of Computer Science, LNCS 324, Springer Verlag, 1988, 280–290.
Diks, K., Djidjev, H. N., Sýkora, O., Vrťo, I., Edge separators for planar graphs with applications, Journal of Algorithms, to appear.
Djidjev, H. N., A liner algorithms for partitioning graphs of fixed genus, SERDICA Bulgaricae Mathematicae Publications, 11, 1985, 369–387.
Djidjev, H. N., A separator theorem, Comptes Rendus de l'Academie Bulgare des Sciences, 34, 1981, 643–645.
Gazit, H., Miller, G. L., Planar separators and Euclidean norm, In:Proc. Algorithms, International Symposium SIGAL'90, 1990, 338–347.
Gilbert, J. R., Hutchinson, J., Tarjan, R. E., A separator theorem for graphs of bounded genus, Journal of Algorithms, 5, 1984, 391–401.
Guy, R. K., Jenkins, T., Schaer, J., The toroidal crossing number of the complete graph, Journal of Combinatorial Theory, 4, 1968, 376–390.
Guy, R. K., Jenkins, T., The toroidal crossing number of the K m,n . Journal of Combinatorial Theory, 6, 1969, 235–250.
Kainen, P. C., On the stable crossing number of cubes, In: Proceedings of the American Mathematical Society, 36, 1972, 55–62.
Leighton, F. T., New lower bound techniques for VLSI, In: Proc. of the 22-nd Annual IEEE Symposium on Foundations of Computer Science, 1981, 1–12.
Leiserson, C. E., Area-efficient graph layouts (for VLSI), In: Proc. of the 21-st Annual IEEE Symposium on Foundations of Computer Science, 1980, 270–280.
Lipton, R. J., Rose, D. J., Tarjan, R. E., Generalized nested dissection, SIAM Journal on Numerical Analysis, 16, 1979, 346–358.
Lipton, R. J., Tarjan, R. E., Application of a planar separator theorem, SIAM Journal on Computing, 9, 1980, 615–627.
Lipton, R. J., Tarjan, R. E., A separator theorem for planar graphs, SIAM Journal on Applied Mathematics, 36, 1979, 177–189.
Miller, G. L., Finding small simple cycle separators for 2-connected planar graphs, Journal of Computer and System Science, 32, 1986, 265–279.
Miller, G. L., Thurston, W., Separators in two and three dimensions, Proc. of the 22-nd Annual ACM Symposium on Theory of Computing, 1990, 300–309.
Mohar, B., Isoperimetric numbers of graphs, Journal of Combinatorial Theory, B, 47, 3, 1989, 274–291.
Monien, B., Sudborough, I. H., Comparing interconnection networks, In: Proc. of the 13-th Symposium on Mathematical Foundations of Computer Science, LNCS 324, Springer Verlag, 1988, 138–153.
Noga, A., Seymour, P., Thomas, R., A separator theorem for graphs with an excluded minor and its applications, In: Proc. of the 22-nd Annual ACM Symposium on Theory of Computing, 1990, 203–299.
Yannakakis, M., Linear and book embedding of graphs, In: Proc. of the Aegean Workshop on Computing, LNCS 227, Springer Verlag, 1986, 229–240.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sýkora, O., Vrto, I. (1992). Edge separators for graphs of bounded genus with applications. In: Schmidt, G., Berghammer, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 1991. Lecture Notes in Computer Science, vol 570. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55121-2_15
Download citation
DOI: https://doi.org/10.1007/3-540-55121-2_15
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-55121-8
Online ISBN: 978-3-540-46735-9
eBook Packages: Springer Book Archive