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).
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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
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DOI: https://doi.org/10.1007/978-88-7642-475-5_47
Publisher Name: Edizioni della Normale, Pisa
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Online ISBN: 978-88-7642-475-5
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