Tree independence number II. Three-path-configurations

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 has a tree decomposition in which every bag has stability number at most c(log⁡n)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.