Max Weight Independent Set in Sparse Graphs with No Long Claws

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, Chudnovsky, Dibek, Rzazewski, 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 graph whose every component is a subdivided claw as an induced subgraph, and a fixed biclique as a subgraph.