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

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 $\text{ker}(G)=\text{core}(G)$, $\text{corona}(G)\cup N(\text{core}(G))=V(G)$ and $\mid \text{corona}(G)\mid +\mid \text{core}(G)\mid =2\alpha (G)+1$ 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: $\text{ker}(G)=\text{core}(G)$ and $\text{corona}(G)\cup N(\text{core}(G))=V(G)$. Moreover, we establish the formula $\mid \text{corona}(G)\mid +\mid \text{core}(G)\mid =2\alpha (G)+k$, 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.