Tree independence number II. Three-path-configurations

A three-paths-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-paths-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 (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 -hard in general, can be solved in quasi-polynomial time if the input graph is 3PC-free.