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

## TL;DR

This paper introduces layered wheel graphs, which have large treewidth and girth, serving as potential examples for characterizing complex graph classes like (theta, triangle)-free and even-hole-free graphs with no K4.

## Contribution

The paper constructs layered wheel graphs demonstrating large treewidth and girth, providing new examples within specific graph classes and insights into their structural properties.

## Key findings

- Layered wheels have arbitrarily large treewidth and girth.
- They are examples within (theta, triangle)-free and even-hole-free graphs with no K4.
- These graphs may help characterize graphs with large treewidth.

## Abstract

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).