(Theta, triangle)-free and (even hole, $K_4$)-free graphs. Part 1 : Layered wheels

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 term of their induced subgraphs (while such a characterization is well understood in term 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~4, a theta is a graph made of three internally vertex disjoint paths of length at least~2 linking two vertices, and $K_4$ is the complete graph on~4 vertices).