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

TL;DR

This paper disproves a previous characterization of graphs with equal independence and annihilation numbers by providing counterexamples and identifying errors, while confirming the theorem for specific graph classes like bipartite and claw-free graphs.

Contribution

The authors disprove the general characterization of graphs with equal independence and annihilation numbers and clarify its validity for certain graph classes.

Findings

Counterexamples with arbitrary parameters are constructed.

The original theorem's proof contains an error.

The theorem holds for bipartite and connected claw-free graphs.

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.