Efficiently recognizing graphs with equal independence and annihilation numbers

TL;DR

This paper proves that recognizing graphs where the independence number equals the annihilation number, or is close to it, can be done efficiently, resolving an open problem in graph theory.

Contribution

It demonstrates that recognizing graphs with independence number equal to or close to the annihilation number is fixed parameter tractable, correcting previous misconceptions.

Findings

Recognition of graphs with α(G)=a(G) is efficient.

Recognition of graphs with α(G)≥a(G)-ℓ is fixed parameter tractable.

The paper corrects prior assumptions about characterization of these graphs.

Abstract

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.