  Let $G$ be a simple graph with vertex set $V(G)$. A set $S\subseteq V(G)$ is
independent if no two vertices from $S$ are adjacent. For $X\subseteq V(G)$,
the difference of $X$ is $d(X) = |X|-|N(X)|$ and an independent set $A$ is
critical if $d(A) = \max \{d(X): X\subseteq V(G) \text{ is an independent
set}\}$ (possibly $A=\emptyset$). Let $\text{nucleus}(G)$ and
$\text{diadem}(G)$ be the intersection and union, respectively, of all maximum
size critical independent sets in $G$. In this paper, we will give two new
characterizations of K\"{o}nig-Egerv\'{a}ry graphs involving
$\text{nucleus}(G)$ and $\text{diadem}(G)$. We also prove a related lower bound
for the independence number of a graph. This work answers several conjectures
posed by Jarden, Levit, and Mandrescu.
