We present a construction called layered wheel. Layered wheels are graphs of arbitrarily large treewidth and girth. They might be an outcome for a possible theorem characterizing graphs with large treewidth in terms of their induced subgraphs (while such a characterization is well-understood in terms of minors). They also provide examples of graphs of large treewidth and large rankwidth in well-studied classes, such as (theta, triangle)-free graphs and even-hole-free graphs with no $K_4$ (where a hole is a chordless cycle of length at least four, a theta is a graph made of three internally vertex disjoint paths of length at least two linking two vertices, and $K_4$ is the complete graph on four vertices).

中文翻译：

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无（Theta，三角形）和（偶数孔，K 4）无图——第 1 部分：分层轮

我们提出了一种称为分层轮的结构。分层轮子是任意大树宽和周长的图。它们可能是根据诱导子图表征具有大树宽的图的可能定理的结果（而这种表征在次要方面很容易理解）。他们还提供了经过充分研究的类中大树宽和大秩宽的图的示例，例如 (theta,triangle)-free 图和没有偶数孔的图${钾}_4$ （其中一个洞是一个长度至少为 4 的无弦循环，一个 theta 是一个由三个内部顶点不相交的路径组成的图，长度至少有两个连接两个顶点，并且 ${钾}_4$ 是四个顶点上的完整图）。