The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, $K_4$)-free graphs was constructed, that still has unbounded tree-width (Sintiari and Trotignon, 2019). The class has unbounded degree and contains arbitrarily large clique-minors. We ask whether this is necessary.

We prove that for every graph $G$, if $G$ excludes a fixed graph $H$ as a minor, then $G$ either has small tree-width, or $G$ contains a large wall or the line graph of a large wall as induced subgraph. This can be seen as a strengthening of Robertson and Seymour’s excluded grid theorem for the case of minor-free graphs. Our theorem implies that every class of even-hole-free graphs excluding a fixed graph as a minor has bounded tree-width. In fact, our theorem applies to a more general class: (theta, prism)-free graphs. This implies the known result that planar even hole-free graphs have bounded tree-width (Silva et al., 2010).

We conjecture that even-hole-free graphs of bounded degree have bounded tree-width. If true, this would mean that even-hole-freeness is testable in the bounded-degree graph model of property testing. We prove the conjecture for subcubic graphs and we give a bound on the tree-width of the class of (even hole, pyramid)-free graphs of degree at most 4.

所有无偶数孔图的类都具有无限的树宽，因为它包含所有完整的图。最近，一类（偶数孔，${钾}_4$)-free 图被构建，它仍然具有无限的树宽（Sintiari 和 Trotignon，2019）。该类具有无限度并包含任意大的小集团。我们问这是否有必要。

我们证明对于每个图 $G$， 如果 $G$ 排除固定图 $H$ 作为未成年人，那么 $G$ 要么树宽小，要么 $G$包含大墙或大墙的线图作为诱导子图。这可以看作是对 Robertson 和 Seymour 的排除网格定理的加强，适用于无次要图的情况。我们的定理意味着，除了作为次要的固定图之外，每类无偶数孔图都具有有界树宽。事实上，我们的定理适用于更一般的类：(theta,prism)-free 图。这意味着已知的结果是平面偶数无孔图具有有界树宽（Silva 等，2010）。

我们推测有界度的偶数无孔图具有有界树宽。如果为真，这将意味着偶数孔自由度在属性测试的有界度图模型中是可测试的。我们证明了亚立方图的猜想，并给出了度数最多为 4 的无（偶数孔、金字塔）图的树宽的界限。