  Let $\alpha(G)$ denote the cardinality of a maximum independent set, while
$\mu(G)$ be the size of a maximum matching in the graph $G=\left(V,E\right) $.
If $\alpha(G)+\mu(G)=\left\vert V\right\vert $, then $G$ is a
K\"onig-Egerv\'ary graph. If $d_{1}\leq d_{2}\leq\cdots\leq d_{n}$ is the
degree sequence of $G$, then the annihilation number $h\left(G\right) $ of $G$
is the largest integer $k$ such that $\sum\limits_{i=1}^{k}d_{i}\leq\left\vert
E\right\vert $ (Pepper 2004, Pepper 2009). A set $A\subseteq V$ satisfying
$\sum \limits_{a\in A} deg(a)\leq\left\vert E\right\vert $ is an annihilation
set, if, in addition, $ deg\left(v\right) +\sum\limits_{a\in A}
deg(a)>\left\vert E\right\vert $, for every vertex $v\in V(G)-A$, then $A$ is a
maximal annihilation set in $G$.
  In (Larson & Pepper 2011) it was conjectured that the following assertions
are equivalent:
  (i) $\alpha\left(G\right) =h\left(G\right) $;
  (ii) $G$ is a K\"onig-Egerv\'ary graph and every maximum independent set is a
maximal annihilating set.
  In this paper, we prove that the implication "(i) $\Longrightarrow$ (ii)" is
correct, while for the opposite direction we provide a series of generic
counterexamples.
  Keywords: maximum independent set, matching, tree, bipartite graph,
K\"onig-Egerv\'ary graph, annihilation set, annihilation number.
