On R-disjoint graphs: a generalization of almost bipartite non-König-Egerváry graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910048081412096 |
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| author | Pereyra, Kevin |
| author_facet | Pereyra, Kevin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09797 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On R-disjoint graphs: a generalization of almost bipartite non-König-Egerváry graphs Pereyra, Kevin Combinatorics 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. |
| title | On R-disjoint graphs: a generalization of almost bipartite non-König-Egerváry graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.09797 |