We prove that the tree-width of graphs in a hereditary class defined by a finite set F of forbidden induced subgraphs is bounded if and only if F includes a complete graph, a complete bipartite graph, a tripod (a forest in which every connected component has at most 3 leaves) and the line graph of a tripod.
@article{arxiv.2012.01115,
title = {Tree-width dichotomy},
author = {Vadim Lozin and Igor Razgon},
journal= {arXiv preprint arXiv:2012.01115},
year = {2021}
}