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Combinatorica

, Volume 28, Issue 1, pp 19–36 | Cite as

Linearity of grid minors in treewidth with applications through bidimensionality

  • Erik D. DemaineEmail author
  • Mohammadtaghi Hajiaghayi
Article

Abstract

We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus grid minors suffice to certify that H-minorfree graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on H-minor-free graphs for any fixed H. Our result has many combinatorial consequences on bidimensionality theory, parameter-treewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial results has several algorithmic consequences including subexponential fixed-parameter algorithms and approximation algorithms.

Mathematics Subject Classification (2000)

05C83 05C85 68R10 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.MITComputer Science and Artificial Intelligence LaboratoryCambridgeUSA

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