Tara Abrishami, Maria Chudnovsky, Marcin Pilipczuk, Paweł Rzążewski

We revisit the recent polynomial-time algorithm for the Max Weight Independent Set (MWIS) problem in bounded-degree graphs excluding a fixed subdivided claw $S_{t,t,t}$ 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 $n^{O(\Delta^2)}$, 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 subdivided claw as an induced subgraph and a fixed biclique as a subgraph.