We show that the Maximum Weight Independent Set problem (MWIS) can be solved in quasi-polynomial time on H-free graphs (graphs excluding a fixed graph H as an induced subgraph) for every H whose every connected component is a path or a subdivided claw (i.e., a tree with at most three leaves). This completes the dichotomy of the complexity of MWIS in F-free graphs for any finite set F of graphs into NP-hard cases and cases solvable in quasi-polynomial time, and corroborates the conjecture that the cases not known to be NP-hard are actually polynomial-time solvable.

The key graph-theoretic ingredient in our result is as follows. Fix an integer t≥1. Let S(t,t,t) be the graph created from three paths on t edges by identifying one endpoint of each path into a single vertex. We show that, given a graph G, one can in polynomial time find either an induced S(t,t,t) in G, or a balanced separator consisting of log(|V(G)|) vertex neighborhoods in G, or an extended strip decomposition of G (a decomposition almost as useful for recursion for MWIS as a partition into connected components) with each particle of weight multiplicatively smaller than the weight of G. This is a strengthening of a result of Majewski et al. [ICALP 2022] which provided such an extended strip decomposition only after the deletion of log(|V(G)|) vertex neighborhoods. To reach the final result, we employ an involved branching strategy that relies on the structural lemma presented above.