# Max Weight Independent Set in sparse graphs with no long claws

Symposium on Theoretical Aspects of Computer Science（2023）

摘要

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, Rzążewski, SODA 2022].
First, we show that with an arguably simpler approach we can obtain a faster
algorithm with running time n^𝒪(Δ^2), where n is the
number of vertices of the instance and Δ 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.

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