For a fixed integer $t \geq 1$, a ($t$-)long claw, denoted $S_{t,t,t}$, is the unique tree with three leaves, each at distance exactly $t$ from the vertex of degree three. Majewski et al. [ICALP 2022, ACM ToCT 2024] proved an analog of the Gy\'{a}rf\'{a}s' path argument for $S_{t,t,t}$-free graphs: given an $n$-vertex $S_{t,t,t}$-free graph, one can delete neighborhoods of $\mathcal{O}(\log n)$ vertices so that the remainder admits an extended strip decomposition (an appropriate generalization of partition into connected components) into particles of multiplicatively smaller size. This statement has proven to be very useful in designing quasi-polynomial time algorithms for Maximum Weight Independent Set and related problems in $S_{t,t,t}$-free graphs. In this work, we refine the argument of Majewski et al. and show that a constant number of neighborhoods suffice.

翻译：对于固定的整数 $t \geq 1$，($t$-)长爪，记为 $S_{t,t,t}$，是唯一具有三个叶子的树，其中每个叶子到度数为三的顶点的距离恰好为 $t$。Majewski 等人 [ICALP 2022, ACM ToCT 2024] 证明了对于不含 $S_{t,t,t}$ 的图，存在一个 Gyárfás 路径论证的类比：给定一个具有 $n$ 个顶点且不含 $S_{t,t,t}$ 的图，我们可以删除 $\mathcal{O}(\log n)$ 个顶点的邻域，使得剩余部分允许一个扩展条带分解（一种划分为连通分量的适当推广）为乘法意义上更小尺寸的粒子。这一结论在设计针对不含 $S_{t,t,t}$ 的图中最大权独立集及相关问题的拟多项式时间算法中已被证明非常有用。在本工作中，我们改进了 Majewski 等人的论证，并证明仅需删除常数个邻域即可达成目标。