Efficiently recognizing graphs with equal independence and annihilation numbers

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 $\alpha (G)\ge a(G)-\mathrm{\ell }$ is fixed parameter tractable using $\mathrm{\ell }$ as parameter.