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径或细分爪（即最多有三个叶子的树）的图$H$，\textsc{最大权独立集}问题（\textsc{MWIS}）可在拟多项式时间内解决$H$-free图（即排除固定图$H$作为诱导子图的图）。这一结果完善了对于任意有限图族$\mathcal{F}$而言，$\mathcal{F}$-free图中\textsc{MWIS}问题复杂性的二分法，将其划分为NP难情形与拟多项式时间可解情形，并佐证了以下猜想：那些尚未被证明为NP难的情形实际上可在多项式时间内求解。我们结果的关键图论基础如下。固定整数$t \geq 1$。设$S_{t,t,t}$为由三条长度为$t$的路径通过将每条路径的一个端点合并为单个顶点而构造的图。我们证明，给定图$G$，可以在多项式时间内找到以下三者之一：$G$中的诱导子图$S_{t,t,t}$；由$G$中$\Oh(\log |V(G)|)$个顶点邻域构成的平衡分隔器；或$G$的扩展带状分解（一种在递归处理\textsc{MWIS}时几乎与连通分量分解同等有用的分解结构），其中每个粒子的权重乘积小于$G$的权重。这一结果强化了Majewski等人[ICALP~2022]的结论，后者仅在删除$\Oh(\log |V(G)|)$个顶点邻域后才提供此类扩展带状分解。为达成最终结果，我们采用了依赖于上述结构引理的复杂分支策略。