Let $\alpha(G)$ denote the cardinality of a maximum independent set, while $\mu(G)$ be the size of a maximum matching in $G=\left( V,E\right) $. It is known that if $\alpha(G)+\mu(G)=\left\vert V\right\vert $, then $G$ is a K\"{o}nig-Egerv\'{a}ry graph. The critical difference $d(G)$ is $\max\{d(I):I\in\mathrm{Ind}(G)\}$, where $\mathrm{Ind}(G)$\ denotes the family of all independent sets of $G$. If $A\in\mathrm{Ind}(G)$ with $d\left( X\right) =d(G)$, then $A$ is a critical independent set. For a graph $G$, let $\mathrm{diadem}(G)=\bigcup\{S:S$ is a critical independent set in $G\}$, and $\varrho_{v}\left( G\right) $ denote the number of vertices $v\in V\left( G\right) $, such that $G-v$ is a K\"{o}nig-Egerv\'{a}ry graph. A graph is called almost bipartite if it has a unique odd cycle. In this paper, we show that if $G$ is an almost bipartite non-K\"{o}nig-Egerv\'{a}ry graph with the unique odd cycle $C$, then the following assertions are true: 1. every maximum matching of $G$ contains $\left\lfloor {V(C)}/{2}\right\rfloor $ edges belonging to $C$; 2. $V(C)\cup N_{G}\left[ \mathrm{diadem}\left( G\right) \right] =V$ and $V(C)\cap N_{G}\left[ \mathrm{diadem}\left( G\right) \right] =\emptyset$; 3. $\varrho_{v}\left( G\right) =\left\vert \mathrm{corona}\left( G\right) \right\vert -\left\vert \mathrm{diadem}\left( G\right) \right\vert $, where $\mathrm{corona}\left( G\right) $ is the union of all maximum independent sets of $G$; 4. $\varrho_{v}\left( G\right) =\left\vert V\right\vert $ if and only if $G=C_{2k+1}$ for some integer $k\geq1$.

翻译：设 $\alpha(G)$ 表示最大独立集的基数，而 $\mu(G)$ 是图 $G=\left( V,E\right) $ 中最大匹配的大小。已知若 $\alpha(G)+\mu(G)=\left\vert V\right\vert $，则 $G$ 是一个König-Egerváry图。临界差 $d(G)$ 定义为 $\max\{d(I):I\in\mathrm{Ind}(G)\}$，其中 $\mathrm{Ind}(G)$ 表示 $G$ 的所有独立集族。若 $A\in\mathrm{Ind}(G)$ 满足 $d\left( X\right) =d(G)$，则 $A$ 是一个临界独立集。对于图 $G$，令 $\mathrm{diadem}(G)=\bigcup\{S:S$ 是 $G$ 中的临界独立集$\}$，并用 $\varrho_{v}\left( G\right) $ 表示满足 $G-v$ 是König-Egerváry图的顶点 $v\in V\left( G\right) $ 的数量。若一个图具有唯一的奇圈，则称其为几乎二部图。本文证明，若 $G$ 是一个具有唯一奇圈 $C$ 的几乎二部非König-Egerváry图，则以下断言成立：1. $G$ 的每个最大匹配均包含 $\left\lfloor {V(C)}/{2}\right\rfloor $ 条属于 $C$ 的边；2. $V(C)\cup N_{G}\left[ \mathrm{diadem}\left( G\right) \right] =V$ 且 $V(C)\cap N_{G}\left[ \mathrm{diadem}\left( G\right) \right] =\emptyset$；3. $\varrho_{v}\left( G\right) =\left\vert \mathrm{corona}\left( G\right) \right\vert -\left\vert \mathrm{diadem}\left( G\right) \right\vert $，其中 $\mathrm{corona}\left( G\right) $ 是 $G$ 的所有最大独立集的并集；4. $\varrho_{v}\left( G\right) =\left\vert V\right\vert $ 当且仅当对某个整数 $k\geq1$ 有 $G=C_{2k+1}$。