  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).
