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) >= a(G) - B is fixed parameter tractable using B as a parameter. (c) 2023 Elsevier B.V. All rights reserved.