Abstract
The Cartesian product C κ × K 2 of a circuit of length κ with K 2 is called a κ-prism. It is well known that graphs not having the κ-prism as a minor have their tree-width bounded by an exponential function of κ. Using brambles and their well-studied relation to tree-width, we show that they have in fact tree-width O(κ 2). As a consequence, we obtain new bounds on the tree-width of graphs having no small grid as a minor.
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
E. Birmelé, J.A. Bondy and B. Reed, The tree-width of the (3 × 3)-grid, manuscript.
R. Diestel, Graph Theory, Second edition. Graduate Texts in Mathematics, 173. Springer-Verlag, New York, 2000.
R. Diestel, K.Yu. Gorbunov, T.R. Jensen and C. Thomassen, Highly connected sets and the excluded grid theorem, J. Combin. Theory Ser.B, 75 (1999), 61–73.
P. Erdős and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470.
B.A. Reed, Tree width and tangles: a new connectivity measure and some applications, in Surveys in Combinatorics, London Math. Soc. Lecture Note Ser. 241, Cambridge Univ. Press, Cambridge, 1997, 87–162.
N. Robertson and P.D. Seymour, Graph minors V: Excluding a planar graph, J. Combin. Theory Ser.B, 41 (1986), 92–114.
P.D. Seymour and R. Thomas, Graph searching and a min-max theorem for tree-width, J. Combin. Theory Ser. B, 58 (1993), 22–33.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Birmelé, E., Bondy, J.A., Reed, B.A. (2006). Brambles, Prisms and Grids. In: Bondy, A., Fonlupt, J., Fouquet, JL., Fournier, JC., Ramírez Alfonsín, J.L. (eds) Graph Theory in Paris. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7400-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7400-6_4
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7228-6
Online ISBN: 978-3-7643-7400-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)