# Linearity of grid minors in treewidth with applications through bidimensionality

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## 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## Preview

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