(Theta, triangle)?free and (even hole, K4)?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 possible theorem characterizing with in terms their induced subgraphs (while such characterization is well-understood minors). also provide examples rankwidth well-studied classes, as (theta, triangle)-free even-hole-free no K 4 (where hole chordless cycle length at least four, theta graph made three internally vertex disjoint paths two linking vertices, the complete on four vertices).