We revisit the recent polynomial-time algorithm for the MAX WEIGHT INDEPENDENT SET (MWIS) problem in bounded-degree graphs that do not contain a fixed graph whose every component is a subdivided claw as an induced subgraph [Abrishami, Dibek, Chudnovsky, Rz\k{a}\.zewski, SODA 2022]. First, we show that with an arguably simpler approach we can obtain a faster algorithm with running time $n^{\mathcal{O}(\Delta^2)}$, where $n$ is the number of vertices of the instance and $\Delta$ is the maximum degree. Then we combine our technique with known results concerning tree decompositions and provide a polynomial-time algorithm for MWIS in graphs excluding a fixed graph whose every component is a subdivided claw as an induced subgraph, and a fixed biclique as a subgraph.

翻译：我们重新审视了近期针对有界度图中最大权重独立集（MWIS）问题的多项式时间算法，其中图不包含每个连通分量均为细分爪的固定图作为导出子图 [Abrishami, Dibek, Chudnovsky, Rz{\k{a}\.zewski, SODA 2022]。首先，我们证明采用一种可论证的更简单方法，可以设计一个运行时间为 $n^{\mathcal{O}(\Delta^2)}$ 的更快算法，其中 $n$ 为实例的顶点数，$\Delta$ 为最大度。随后，我们将该技术结合已知的树分解相关理论，给出一个多项式时间算法，适用于不含每个连通分量均为细分爪的固定图作为导出子图，且不含固定完全二分图作为子图的图类中的MWIS问题。