Max Weight Independent Set in sparse graphs with no long claws

Abstract

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, RzÄ Å¼ewski, SODA 2022]. First, we show that with an arguably simpler approach we can obtain a faster algorithm with running time $n^{\mathcal{O}(\hat{I}{”}^2)}$, where $n$ is the number of vertices of the instance and $\hat{I}”$ 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.

arXiv admin note: text overlap with arXiv:2107.05434