A three-path-configuration is a graph consisting of three pairwise internally-disjoint paths the union of every two of which is an induced cycle of length at least four. A graph is 3PC-free if no induced subgraph of it is a three-path-configuration. We prove that 3PC-free graphs have poly-logarithmic tree-independence number. More explicitly, we show that there exists a constant c such that every n-vertex 3PC-free graph graph has a tree decomposition in which every bag has stability number at most c(logn)2. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems, that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is 3PC-free.
@article{arxiv.2405.00265,
title = {Tree independence number II. Three-path-configurations},
author = {Maria Chudnovsky and Sepehr Hajebi and Daniel Lokshtanov and Sophie Spirkl},
journal= {arXiv preprint arXiv:2405.00265},
year = {2025}
}