Abstract
Bounded degree trees and bounded degree planar graphs on n vertices are known to admit bisections of width O(log n) and \( O(\sqrt n ) \), respectively. We investigate the structure of graphs that meet this bound. In particular, we show that such a tree must have diameter O(n/log n) and such a planar graph must have tree width \( O(\sqrt n ) \). To show the result for trees, we derive an inequality that relates the width of a minimum bisection with the diameter of a tree.
The first author was partially supported by CNPq Proc. 308523/2012-1 and 477203/2012-4, and Project MaCLinC of NUMEC/USP. The second author gratefully acknowledges the support by the Evangelische Studienwerk Villigst e.V. and was partially supported by TopMath, the graduate program of the Elite Network of Bavaria and the graduate center of TUM Graduate School. The third author was supported by DFG grant TA 309/2-2. The cooperation of the three authors was supported by a joint CAPES-DAAD project (415/ppp-probral/po/D08/11629, Proj. no. 333/09).
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
S. Arora, D. Karger and M. Karpinski, Polynomial time approximation schemes for dense instances of NP-hard problems, Journal of Computer and System Sciences 58(1) (1999), 193–210.
N. Alon, P. Seymour and R. Thomas, Planar separators, SIAM Journal on Discrete Mathematics 7(2) (1994), 184–193.
P. Berman and M. Karpinski, Approximation hardness of bounded degree MIN-CSP and MIN-BISECTION, In: “Automata, Languages and Programming”, vol. 2380 of LNCS, Springer, Berlin, 2002, 623–632.
M. Garey, D. Johnson and L. Stockmeyer, Some simplified NP-complete graph problems, Theoretical Computer Science 1(3) (1976), 237–267.
K. Jansen et al., Polynomial time approximation schemes for maxbisection on planar and geometric graphs, SIAM Journal on Computing 35(1) (2005), 110–119.
H. Räcke, Optimal hierarchical decompositions for congestion minimization in networks, In: “Proceedings of the 40th annual ACM symposium on Theory of computing”, STOC’ 08, New York, NY, USA, 2008, ACM, 255–264.
N. Robertson, P. Seymour and R. Thomas, Quickly excluding a planar graph, Journal of Combinatorial Theory, Series B 62(2) (1994), 323–348.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2013 Scuola Normale Superiore Pisa
About this paper
Cite this paper
Fernandes, C.G., Schmidt, T.J., Taraz, A. (2013). On the structure of graphs with large minimum bisection. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_47
Download citation
DOI: https://doi.org/10.1007/978-88-7642-475-5_47
Publisher Name: Edizioni della Normale, Pisa
Print ISBN: 978-88-7642-474-8
Online ISBN: 978-88-7642-475-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)