Very recently, Thomass\'e, Trotignon and Vuskovic [WG 2014] have given an FPT
algorithm for Weighted Independent Set in bull-free graphs parameterized by the
weight of the solution, running in time 2O(k5)⋅n9. In this article
we improve this running time to 2O(k2)⋅n7. As a byproduct, we also
improve the previous Turing-kernel for this problem from O(k5) to O(k2).
Furthermore, for the subclass of bull-free graphs without holes of length at
most 2p−1 for p≥3, we speed up the running time to 2O(k⋅kp−11)⋅n7. As p grows, this running time is
asymptotically tight in terms of k, since we prove that for each integer p≥3, Weighted Independent Set cannot be solved in time 2o(k)⋅nO(1) in the class of {bull,C4,…,C2p−1}-free graphs unless the
ETH fails.Comment: 15 page