  We show that the \textsc{Maximum Weight Independent Set} problem
(\textsc{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
\textsc{MWIS} in $\mathcal{F}$-free graphs for any finite set $\mathcal{F}$ 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 $t \geq 1$. 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 $\Oh(\log |V(G)|)$
vertex neighborhoods in $G$, or an extended strip decomposition of $G$ (a
decomposition almost as useful for recursion for \textsc{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 $\Oh(\log |V(G)|)$ vertex neighborhoods. To reach the
final result, we employ an involved branching strategy that relies on the
structural lemma presented above.
