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：我们重新审视了针对有界度图中最大权重独立集（MWIS）问题的最新多项式时间算法，其中图不包含每个连通分支均为细分爪的固定图作为诱导子图 [Abrishami, Dibek, Chudnovsky, Rz\k{a}\.zewski, SODA 2022]。首先，我们证明通过一种更简单的方法可以获得运行时间为 $n^{\mathcal{O}(\Delta^2)}$ 的更快算法，其中 $n$ 是实例的顶点数，$\Delta$ 是最大度。然后，我们将我们的技术与已知的树分解结果相结合，为不含每个连通分支均为细分爪的固定图作为诱导子图、且不含固定双团作为子图的图提供了MWIS的多项式时间算法。