On Minimal Critical Independent Sets of Almost Bipartite non-Konig-Egervary Graphs

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 $d\left(G\right)=\max \{\mid A\mid -\mid N\left(A\right)\mid :A\subseteq V\}$ is called the critical difference of $G$ (where $N\left(A\right)=\{v:v\in V,N\left(v\right)\cap A\mathrm{\ne }\mathrm{\varnothing }\}$). It is known that $\alpha (G)-\mu (G)\le d\left(G\right)$ 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 $\ker (G)=\bigcap \{S:S\text{ is a critical independent set}\}$, $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}\left(G\right)$ be the intersection of all maximum independent sets, and $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{a}\left(G\right)$ be the union of all maximum independent sets of $G$. It is known that $\ker (G)\subseteq \mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(G)$ 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 $\ker (G)=$ $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(G)$, $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{a}(G)$ $\cup$ $N(\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}\left(G\right))=V(G)$, and $\mid \mathrm{c}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{a}(G)\mid +\mid \mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(G)\mid =2\alpha (G)+1$.