Let $α(G)$ denote the cardinality of a maximum independent set, while $μ(G)$ be the size of a maximum matching in $G=\left( V,E\right) $. Let $ξ(G)$ denote the size of the intersection of all maximum independent sets. It is known that if $α(G)+μ(G)=n(G)=\left\vert V\right\vert $, then $G$ is a König-Egerváry graph. If $α(G)+μ(G)=n(G) -1$, then $G$ is a $1$-König-Egerváry graph. If $G$ is not a König-Egerváry graph, and there exists a vertex $v\in V$ (an edge $e\in E$) such that $G-v$ ($G-e$) is König-Egerváry, then $G$ is called a vertex (an edge) almost König-Egerváry graph (respectively). 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. Let $diadem (G)=\bigcup\{S:S$ 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 König-Egerváry graph. In this paper, we characterize all types of almost König-Egerváry graphs and present interrelationships between them. We also show that if $G$ is a $1$-König-Egerváry graph, then $\varrho_{v}\left( G\right) \leq n\left( G\right) +d\left( G\right) -ξ\left( G\right) -β(G)$, where $β(G)=\left\vert diadem(G)\right\vert $. As an application, we characterize the $1$-König-Egerváry graphs that become König-Egerváry after deleting any vertex.