A generalized t-pyramid is a graph obtained from a certain kind of tree (a subdivided star or a subdivided cubic caterpillar) and the line graph of a subdivided cubic caterpillar by identifying simplicial vertices. We prove that for every integer t there exists a constant c(t) such that every n-vertex even-hole-free graph with no clique of size t and no induced subgraph isomorphic to a generalized t-pyramid has treewidth at most c(t)logn. This settles a special case of a conjecture of Sintiari and Trotignon; this bound is also best possible for the class. It follows that several \textsf{NP}-hard problems such as \textsc{Stable Set}, \textsc{Vertex Cover}, \textsc{Dominating Set} and \textsc{Coloring} admit polynomial-time algorithms on this class of graphs. Results from this paper are also used in later papers of the series, in particular to solve the full version of the Sintiari-Trotignon conjecture.
@article{arxiv.2307.13684,
title = {Induced subgraphs and tree decompositions X. Towards logarithmic treewidth for even-hole-free graphs},
author = {Tara Abrishami and Bogdan Alecu and Maria Chudnovsky and Sepehr Hajebi and Sophie Spirkl},
journal= {arXiv preprint arXiv:2307.13684},
year = {2025}
}