  The independence number $\alpha(G)$ is the cardinality of a maximum
independent set, while $\mu(G)$ is the size of a maximum matching in $G$. If
$\alpha(G)+\mu(G)$ equals the order of $G$, then $G$ is called a Konig-Egervary
graph. The number $d\left( G\right) =\max\{\left\vert A\right\vert -\left\vert
N\left( A\right) \right\vert :A\subseteq V\}$ is called the critical difference
of $G$ (where $N\left( A\right) =\left\{ v:v\in V,N\left( v\right) \cap
A\neq\emptyset\right\} $). It is known that $\alpha(G)-\mu(G)\leq d\left(
G\right) $ holds for every graph. A graph $G$ is unicyclic if it has a unique
cycle and almost bipartite if it has only one odd cycle. Let
$\mathrm{\ker}(G)=\bigcap\left\{ S:S\text{ is a critical independent
set}\right\} $, $\mathrm{core}\left( G\right) $ be the intersection of all
maximum independent sets, and $\mathrm{corona}\left( G\right) $ be the union of
all maximum independent sets of $G$. It is known that
$\mathrm{\ker}(G)\subseteq\mathrm{core}(G)$ is true for every graph, while the
equality holds for bipartite graphs, and for unicyclic non-Konig-Egervary
graphs. In this paper, we prove that if $G$ is an almost bipartite
non-Konig-Egervary graph, then $\mathrm{\ker}(G)=$ $\mathrm{core}(G)$,
$\mathrm{corona}(G)$ $\cup$ $N(\mathrm{core} \left( G\right) )=V(G)$, and
$\left\vert \mathrm{corona}(G)\right\vert +\left\vert
\mathrm{core}(G)\right\vert =2\alpha(G)+1$.
