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.

翻译：我们证明：对于每个连通分量均为路或细分爪（即最多三片叶子的树）的图 $H$，\textsc{最大权独立集}问题（\textsc{MWIS}）可在拟多项式时间内于 $H$-free图（不含固定图 $H$ 作为导出子图的图）上求解。这一结果完善了任意有限图族 $\mathcal{F}$ 的 $\mathcal{F}$-free图中 \textsc{MWIS} 复杂度二分性（分为NP难情形与拟多项式时间可解情形），并佐证了如下猜想：尚未被证明为NP难的情形实际上可在多项式时间内求解。我们结果的核心图论工具如下：固定整数 $t \geq 1$。令 $S_{t,t,t}$ 为将三条 $t$ 边路径的端点分别粘合至同一顶点所生成的图。我们证明：给定图 $G$，可在多项式时间内找到 $G$ 中的导出子图 $S_{t,t,t}$，或由 $\Oh(\log |V(G)|)$ 个顶点邻域构成的平衡分离器，或 $G$ 的扩展条带分解（一种对 \textsc{MWIS} 递归处理几乎与连通分量分解同样有用的结构），其中每个粒子的权重乘积极小于 $G$ 的权重。这一结果强化了 Majewski 等人 [ICALP~2022] 的结论——后者仅能在删除 $\Oh(\log |V(G)|)$ 个顶点邻域后提供此类扩展条带分解。为得到最终结果，我们采用了依赖上述结构引理的复杂分支策略。