  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 $2^{O(k^5)} \cdot n^9$. In this article
we improve this running time to $2^{O(k^2)} \cdot n^7$. As a byproduct, we also
improve the previous Turing-kernel for this problem from $O(k^5)$ to $O(k^2)$.
Furthermore, for the subclass of bull-free graphs without holes of length at
most $2p-1$ for $p \geq 3$, we speed up the running time to $2^{O(k \cdot
k^{\frac{1}{p-1}})} \cdot n^7$. As $p$ grows, this running time is
asymptotically tight in terms of $k$, since we prove that for each integer $p
\geq 3$, Weighted Independent Set cannot be solved in time $2^{o(k)} \cdot
n^{O(1)}$ in the class of $\{bull,C_4,\ldots,C_{2p-1}\}$-free graphs unless the
ETH fails.
