This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of the $(k\times k)$-wall or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows.

• 1.

  For $t\ge 2$, a t-theta is a graph consisting of two nonadjacent vertices and three internally vertex-disjoint paths between them, each of length at least t. A t-pyramid is a graph consisting of a vertex v, a triangle B disjoint from v and three paths starting at v and vertex-disjoint otherwise, each joining v to a vertex of B, and each of length at least t. We prove that for all $k,t$ and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a t-theta, or a t-pyramid, or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a t-theta for some $t\ge 2$).

• 2.

  A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every Δ and subcubic subdivided caterpillar T, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph.

本文的动机是以下问题：大树宽图不可避免的导出子图是什么？阿布尔克等人。做出了一个猜想，在有界最大度图中回答了这个问题，断言对于所有k和 Δ，每个具有最大度至多 Δ 且树宽足够大的图都包含$（k\times k）$-墙或细分的线图$（k\times k）$-wall 作为诱导子图。我们证明了支持这个猜想的两个定理，如下。

• 1.

  为了$t\ge 2$，t-theta是由两个不相邻的顶点和它们之间的三个内部顶点不相交路径组成的图，每个路径的长度至少为t。t金字塔是由顶点v 、与v不相交的三角形B以及从v开始且顶点不相交的三个路径组成的图，每条路径将v连接到B的顶点，并且每条路径的长度至少为t。我们证明对于所有人$k,t$和 Δ，每个具有最大度数至多 Δ 且树宽足够大的图包含t -theta，或t -金字塔，或细分的线图$（k\times k）$-wall 作为诱导子图。这肯定地回答了 Pilipczuk 等人的问题。询问每个有界最大度和足够大的树宽的图是否包含 θ 或三角形作为导出子图（其中 θ对于某些来说意味着t -theta$t\ge 2$）。

• 2.

  次立方细分毛毛虫是一棵最大度数最多为三的树，其所有三度顶点都位于一条路径上。我们证明，对于每个 Δ 和亚立方细分毛毛虫T ，每个最大度数为 Δ 且树宽足够大的图都包含T的细分或T细分的线图作为诱导子图。