  A set $S\subseteq V$ is \textit{independent} in a graph $G=\left( V,E\right)
$ if no two vertices from $S$ are adjacent. The \textit{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 \textit{K\"{o}nig-Egerv\'{a}ry graph
}\cite{dem,ster}. 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
\textit{critical difference} of $G$ \cite{Zhang} (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
\cite{Levman2011a,Lorentzen1966,Schrijver2003}. In \cite{LevMan5} it was shown
that $d(G)=\alpha(G)-\mu(G)$ is true for every K\"{o}nig-Egerv\'{a}ry graph.
  A graph $G$ is \textit{(i)} \textit{unicyclic} if it has a unique cycle,
\textit{(ii)} \textit{almost bipartite} if it has only one odd cycle. It was
conjectured in \cite{LevMan2012a,LevMan2013a} and validated in
\cite{Bhattacharya2018} that $d(G)=\alpha(G)-\mu(G)$ holds for every unicyclic
non-K\"{o}nig-Egerv\'{a}ry graph $G$.
  In this paper we prove that if $G$ is an almost bipartite graph of order
$n\left( G\right) $, then $\alpha(G)+\mu(G)\in\left\{ n\left( G\right)
-1,n\left( G\right) \right\} $. Moreover, for each of these two values, we
characterize the corresponding graphs. Further, using these findings, we show
that the critical difference of an almost bipartite graph $G$ satisfies \[
d(G)=\alpha(G)-\mu(G)=\left\vert \mathrm{core}(G)\right\vert -\left\vert
N(\mathrm{core}(G))\right\vert , \] where by \textrm{core}$\left( G\right) $ we
mean the intersection of all maximum independent sets.
