Linear min-max relation between the treewidth of H-minor-free graphs and its largest grid

Abstract

A key theorem in algorithmic graph-minor theory is a min-max relation
between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson
and Seymour, which ultimately proves Wagner's Conjecture about the structure of minor-closed graph properties.

In 2008, Demaine and Hajiaghayi proved a remarkable linear min-max
relation for graphs excluding any fixed minor H: every H-minor-free
graph of treewidth at least c_H r has an r times r-grid minor for some
constant c_H. However, as they pointed out, there is still a major
problem left in this theorem.  The problem is that their proof heavily
depends on Graph Minor Theory, most of which lacks explicit bounds and
is believed to have very large bounds. Hence c_H  is not explicitly
given in the paper and therefore this result is usually not strong
enough to derive efficient algorithms.

Motivated by this problem,  we give  another (relatively short  and
simple) proof of this result without using big machinery of Graph
Minor Theory. Hence we can give an explicit bound for c_H (an exponential function of a polynomial of |H|). Furthermore, our result
gives a constant w=2^O(r^2 log r) such that every graph of treewidth
at least w has an r times r-grid minor, which improves the previously
known best bound 2^Theta(r^5)$ given by Robertson, Seymour, and Thomas
in 1994.