  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 $\Delta$, every graph with maximum degree at
most $\Delta$ 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 $t\geq 2$, 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 $\Delta$, every graph with maximum degree at most $\Delta$
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 $t\geq 2$).
  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
$\Delta$ and subcubic subdivided caterpillar $T$, every graph with maximum
degree at most $\Delta$ and sufficiently large treewidth contains either a
subdivision of $T$ or the line graph of a subdivision of $T$ as an induced
subgraph.
