Core and Corona in 2-Bicritical Odd-Bicyclic Graphs

TL;DR

This paper classifies the structure of 2-bicritical graphs with up to two odd cycles, explicitly computes their maximum independent sets, core, and corona, and extends existing theories to a broader class of graphs.

Contribution

It provides a complete structural classification of 2-bicritical graphs with at most two odd cycles, including explicit calculations and characterizations.

Findings

Explicit formulas for α(G), core G, and corona G for each graph family.

The sum of the sizes of core G and corona G is 2α(G), 2α(G)+1, or 2α(G)+2.

Structural characterization based on the position of odd cycles.

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 $\alpha(G)$, $\core G$, and $\corona G$, and describe the corresponding matching structure. We prove that $\a{\core G}+\a{\corona G}$ 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 K\"onig--Egerv\'ary graphs and later for almost bipartite graphs to a broader non-K\"onig--Egerv\'ary setting.