Counterexamples to the characterisation of graphs with equal independence and annihilation number

Abstract

We disprove the characterisation of graphs with equal independence and annihilation number by Larson and Pepper (2011). Series of counterexamples with arbitrary number of vertices, arbitrary number of components, arbitrary large independence number and arbitrary large difference between the critical and the regular independence number are provided. Furthermore, we point out the error in the proof of the theorem. However, we show that the theorem still holds for bipartite graphs and connected claw-free graphs.