Core and Corona in 2-Bicritical Odd-Bicyclic Graphs

Let $\text{core}G$ and $\text{corona}G$ denote the intersection and the union, respectively, of all maximum independent sets of a graph $G$. A graph is called \emph{$2$-bicritical} if $\text{a}N(S)>\text{a}S$ for every nonempty independent set $S$. Pulleyblank 1979 showed that almost all graphs are $2$-bicritical. In this paper, we study the structure of maximum independent sets in $2$-bicritical graphs with at most two odd cycles. Using ear–pendant decompositions, we obtain a complete structural classification of these graphs into four families: one-odd cycle, fused-odd, even-linked, and odd-linked graphs. For each family, we compute explicitly $\alpha (G)$, $\text{core}G$, and $\text{corona}G$, and describe the corresponding matching structure. We prove that $\text{a}\text{core}G+\text{a}\text{corona}G$ equals either $2\alpha (G),2\alpha (G)+1$ or $2\alpha (G)+2$, and we give a complete, purely structural characterization of the graphs in each case in terms of the relative position of their odd cycles. These results extend a theory originally developed for König–Egerváry graphs and later for almost bipartite graphs to a broader non-König–Egerváry setting.