  What are the unavoidable induced subgraphs of graphs with large treewidth? It
is well-known that the answer must include a complete graph, a complete
bipartite graph, all subdivisions of a wall and line graphs of all subdivisions
of a wall (we refer to these graphs as the "basic treewidth obstructions"). So
it is natural to ask whether graphs excluding the basic treewidth obstructions
as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered
this question in the negative. Their counterexamples, the so-called "layered
wheels," contain wheels, where a wheel consists of a hole (i.e., an induced
cycle of length at least four) along with a vertex with at least three
neighbors in the hole. This leads one to ask whether graphs excluding wheels
and the basic treewidth obstructions as induced subgraphs have bounded
treewidth. This also turns out to be false due to Davies' recent example of
graphs with large treewidth, no wheels and and no basic treewidth obstructions
as induced subgraphs. However, in Davies' example there exist holes and
vertices (outside of the hole) with two neighbors in them. Here we prove that a
hole with a vertex with at least two neighbors in it is inevitable in graphs
with large treewidth and no basic obstruction. Our main result is that graphs
in which every vertex has at most one neighbor in every hole (that does not
contain it) and with the basic treewidth obstructions excluded as induced
subgraphs have bounded treewidth.
