Core and Corona in 2-Bicritical Odd-Bicyclic Graphs

Abstract

Let $\core G$ and $\corona G$ denote the intersection and the union, respectively, of all maximum independent sets of a graph $G$. A graph is called \emph{$2$-bicritical} if $\a{N(S)}>\a S$ 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 $α(G)$, $\core G$, and $\corona G$, and describe the corresponding matching structure. We prove that $\a{\core G}+\a{\corona G}$ equals either $2α(G),2α(G)+1$ or $2α(G)+2$, 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.

Figures (7)

• Figure 1: Examples of one-odd cycle and fused-odd graphs.
• Figure 2: Example of even-linked and odd-linked graphs.
• Figure 3: Improper addition of an odd ear produces additional odd cycles: three representative configurations.
• Figure 4: An illustration of the proof of \ref{['oi12j3j123ijj']}
• Figure 5: An illustration of the proof of \ref{['12oij3']}

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2: levit2025almost
• Theorem 2.1: edmonds1965pathsgallai1964maximale Gallai--Edmonds structure theorem
• Theorem 3.1: pulleyblank1979minimum
• Theorem 3.2: bourjolly1989konig
• Theorem 3.3: lovasz1983ear
• Proposition 3.4
• proof
• Theorem 4.1: berge2005some
• Theorem 4.2