  The annihilation number $a(G)$ of a graph $G$ is an efficiently computable
upper bound on the independence number $\alpha(G)$ of $G$. Recently, Hiller
observed that a characterization of the graphs $G$ with $\alpha(G)=a(G)$ due to
Larson and Pepper is false. Since the known efficient algorithm recognizing
these graphs was based on this characterization, the complexity of recognizing
graphs $G$ with $\alpha(G)=a(G)$ was once again open. We show that these graphs
can indeed be recognized efficiently. More generally, we show that recognizing
graphs $G$ with $\alpha(G)\geq a(G)-\ell$ is fixed parameter tractable using
$\ell$ as parameter.
