A characterization of graphs with $\a{\corona G}+\a{\core G}=2\alpha(G)+1$

A K\H{o}nig--Egerv\'ary graph is a graph $G$ satisfying $\alpha(G)+\mu(G)=n(G)$, where $\alpha(G)$, $\mu(G)$, and $n(G)$ denote the independence number, the matching number, and the order of $G$, respectively. Let $\textnormal{core}(G)$ and $\textnormal{corona}(G)$ be the intersection and the union of all maximum independent sets of $G$. In this paper, we provide a complete characterization of graphs satisfying $\a{\corona G}+\a{\core G}=2\alpha(G)+1$, thus giving a solution to an open problem posed by Levit and Mandrescu. It is known that for a non-K\H{o}nig--Egerv\'ary graph with a unique odd cycle, the following hold: $\ker G=\textnormal{core}(G),\allowbreak\ \left|\textnormal{corona}(G)\right| +\left|\textnormal{core}(G)\right| =2\alpha(G)+1,\allowbreak\ \textnormal{corona}(G)\cup N(\textnormal{core}(G))=V(G)$. We extend these three results to a family of graphs containing an arbitrarily large number of odd cycles.