A clock is a graph consisting of an induced cycle C and a vertex not in C with at least two non-adjacent neighbours in C. We show that every clock-free graph of large treewidth contains a "basic obstruction" of large treewidth as an induced subgraph: a complete graph, a subdivision of a wall, or the line graph of a subdivision of a wall.
@article{arxiv.2311.05719,
title = {Induced subgraphs and tree decompositions XIV. Non-adjacent neighbours in a hole},
author = {Maria Chudnovsky and Sepehr Hajebi and Sophie Spirkl},
journal= {arXiv preprint arXiv:2311.05719},
year = {2025}
}