On R-disjoint graphs: a generalization of almost bipartite non-König-Egerváry graphs

Abstract

An almost bipartite graph is a graph with a unique odd cycle. Levit and Mandrescu showed that in every non-König--Egerváry almost bipartite graph the equalities $\textnormal{ker}(G)=\textnormal{core}(G)$, $\textnormal{corona}(G)\cup N(\textnormal{core}(G)) = V(G)$ and $\left|\textnormal{corona}(G)\right|+\left|\textnormal{core}(G)\right|=2α(G)+1$ hold. In this work, we present a generalization of this theory by introducing the family of $R$-disjoint graphs, which contains all non-König--Egerváry almost bipartite graphs, allowing the presence of multiple odd cycles under connectivity constraints based on the reach sets $R(C)$. We prove that $R$-disjoint graphs preserve the fundamental properties of almost bipartite graphs: $\textnormal{ker}(G)=\textnormal{core}(G)$ and $\textnormal{corona}(G)\cup N(\textnormal{core}(G))=V(G)$. Moreover, we establish the formula $\left|\textnormal{corona}(G)\right|+\left|\textnormal{core}(G)\right|=2α(G)+k$, where $k$ is the number of disjoint odd cycles in $G$, which refines the previously known particular case when $k=1$. $R$-disjoint graphs naturally induce a canonical decomposition; we obtain structural properties of this decomposition and, as a consequence, verify a recent conjecture of Levit and Mandrescu.