On R-disjoint graphs: a generalization of almost bipartite non-K\"onig-Egerv\'ary graphs

TL;DR

This paper introduces R-disjoint graphs, a generalization of almost bipartite graphs with multiple odd cycles, preserving key properties and refining known formulas related to graph invariants.

Contribution

It extends the theory of almost bipartite graphs to R-disjoint graphs, allowing multiple odd cycles while maintaining core properties and providing a new formula involving the number of odd cycles.

Findings

R-disjoint graphs preserve key properties of almost bipartite graphs

Established a formula relating corona, core, and independence number with odd cycles

Verified a recent conjecture of Levit and Mandrescu

Abstract

An almost bipartite graph is a graph with a unique odd cycle. Levit and Mandrescu showed that in every non-K\"onig--Egerv\'ary almost bipartite graph the equalities $\textnormal{ker}(G)=\textnormal{core}(G)$, $\textnormal{corona}(G)\cup N(\textnormal{core}(G)) = V(G)$ and $\left|\textnormal{corona}(G)\right|+\left|\textnormal{core}(G)\right|=2\alpha(G)+1$ hold. In this work, we present a generalization of this theory by introducing the family of $R$-disjoint graphs, which contains all non-K\"onig--Egerv\'ary 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 $\left|\textnormal{corona}(G)\right|+\left|\textnormal{core}(G)\right|=2\alpha(G)+k$, 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.