Linearity of grid minors in treewidth with applications through bidimensionality

We prove that any H -minor-free graph, for a fixed graph H , of treewidth w has an ( w ) ( 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.