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

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.