Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph has treewidth k if it can be decomposed by a sequence of noncrossing cutsets of size at most k into pieces of size at most $k+1$. The study of hereditary graph classes (i.e. those closed under vertex deletion only) reveals a different picture, where cutsets that are not necessarily bounded in size (such as star cutsets, 2-joins and their generalization) are required to decompose the graph into simpler pieces that are structured but not necessarily bounded in size. A number of such decomposition theorems are known for complex hereditary graph classes, including even-hole-free graphs, perfect graphs and others. These theorems do not describe the global structure in the sense that a tree decomposition does, since the cutsets guaranteed by them are far from being noncrossing. They are also of limited use in algorithmic applications.

We show that in the case of even-hole-free graphs of bounded degree the cutsets described in the previous paragraph can be partitioned into a bounded number of well-behaved collections. This allows us to prove that even-hole-free graphs with bounded degree have bounded treewidth, resolving a conjecture of Aboulker et al. (2021) [1]. As a consequence, it follows that many algorithmic problems can be solved in polynomial time for this class, and that even-hole-freeness is testable in the bounded degree graph model of property testing. In fact we prove our results for a larger class of graphs, namely the class of $C_4$-free odd-signable graphs with bounded degree.

Treewidth 是从对图的次要封闭类（即在顶点和边删除以及边收缩下封闭的类）的研究中出现的参数。它在某种意义上描述了图的全局结构。粗略地说，一个图的树宽为k，如果它可以被一系列大小至多为k的非交叉割集分解为最多大小为$ķ+1$. 对遗传图类（即仅在顶点删除下封闭的图类）的研究揭示了另一幅图，其中需要大小不一定有界的割集（例如星割集、2-join 及其泛化）将图分解为更简单的图结构化但不一定有大小限制的片段。许多这样的分解定理以复杂的遗传图类而闻名，包括偶数无孔图、完美图等。这些定理并没有像树分解那样描述全局结构，因为它们所保证的割集远非不交叉。它们在算法应用中的用途也有限。

我们表明，在有界度的偶数无孔图的情况下，上一段中描述的割集可以划分为有界数量的行为良好的集合。这使我们能够证明具有有界度的偶数无孔图具有有界树宽，从而解决了 Aboulker 等人的猜想。(2021) [1] 。因此，该类的许多算法问题可以在多项式时间内解决，并且偶数无孔在属性测试的有界度图模型中是可测试的。事实上，我们为更大的图类证明了我们的结果，即$C_4$- 有界度的无奇数可签名图。