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 ∆, every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a subdivision of the (k × k)-wall or the line graph of a subdivision of the (k × k)-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows.1. For t ≥ 2, a t-theta is a graph consisting of two nonadjacent vertices and three internally 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 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 ∆, every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a t-theta, or a t-pyramid, or the line graph of a subdivision of the (k × 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 ≥ 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 ∆ and subcubic subdivided caterpillar T , every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph.

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A graph is strongly perfect if every induced subgraph H of it has a stable set that meets every maximal clique of H. A graph is claw‐free if no vertex has three pairwise nonadjacent neighbors. The characterization of claw‐free graphs that are strongly perfect by a set of forbidden induced subgraphs was conjectured by Ravindra in 1990 and was proved by Wang in 2006. Here we give a shorter proof of this characterization.