A key theorem in algorithmic graph-minor theory is a min-max relation between the treewidth of a graph (i.e., the minimum width of a tree-decomposition) and the maximum size of a 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 properties of minor-closed graphs.

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, a major issue with this theorem is that their proof heavily depends on the graph minor theory, most of which lacks explicit bounds and is believed to have very large bounds.

Motivated by this problem, we give another (relatively short and simple) proof of this result without using the machinery of the graph minor theory. Hence we give an explicit bound for $c_H$, which is an exponential function of a polynomial in $|H|$. 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.

算法图次要理论中的一个关键定理是图的树宽（即，树分解的最小宽度）与网格小图的最大大小之间的最小-最大关系。这个最小-最大关系是罗伯逊和西摩的图次要理论的基石，该理论最终证明了瓦格纳关于次要闭合图的性质的猜想。

Demaine和Hajiaghayi对于不包含任何固定次要H的图证明了显着的线性min-max关系：每个树宽的无H小图至少$C_H\mathrm{[R}$ 有一个 $\mathrm{[R}\times \mathrm{[R}$-grid未成年人的一些常数 $C_H$。但是，正如他们所指出的那样，该定理的一个主要问题是，他们的证明在很大程度上取决于图的次要理论，其中大多数都没有明确的界限，并且据信具有很大的界限。

受此问题的影响，我们给出了另一种（相对简短和简单）的结果证明，而没有使用图次要理论的原理。因此，我们为$C_H$，这是多项式中的指数函数 $|H|$。此外，我们的结果给出了一个常数$w=2^{Ø（{\mathrm{[R}}^2\mathrm{日志}\mathrm{[R}）}$这样，每个至少w的树宽图都有一个$\mathrm{[R}\times \mathrm{[R}$-电网未成年人。