Solving the Weighted Stable Set Problem in Claw-Free Graphs via Decomposition

We propose an algorithm for solving the maximum weighted stable set problem on claw-free graphs that runs in O (| V |(| E | + | V | log| V |))-time, drastically improving the previous best known complexity bound. This algorithm is based on a novel decomposition theorem for claw-free graphs, which is also introduced in the present article. Despite being weaker than the structural results for claw-free graphs given by Chudnovsky and Seymour [2005, 2008a, 2008b] our decomposition theorem is, on the other hand, algorithmic, that is, it is coupled with an O (| V || E |)-time algorithm that actually produces the decomposition.