Let α(G) denote the cardinality of a maximum independent set, while μ(G) be the size of a maximum matching in graph G = (V,E). If α(G)+μ(G) = |V |, then G is a K¨onig-Egerv´ary graph, while if α(G) + μ(G) = |V | − k, then G is a k-K¨onig-Egerv´ary graph. 1-K¨onig-Egerv´ary graphs are also known as almost K¨onig-Egerv´ary graphs. If G is not K¨onig-Egerv´ary, but there exists a vertex v ∈ V (an edge e ∈ E) such that G−v (G−e) is K¨onig-Egerv´ary, then G is called a vertex almost K¨onig-Egerv´ary (an edge almost K¨onig-Egerv´ary graph, respectively). The presentation mostly concerns several interrelations between structural properties of almost (vertex / edge) K¨onig-Egerv´ary graphs.