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.

Figures (5)

• Figure 1: In this figure, we show a graph $G$ of order $18$. A maximum matching is highlighted in red, and a maximum independent set in blue. Notice that $\alpha(G)=9$, while $\textnormal{corona}(G)=V(G)\setminus\{4,16\}$, and $\textnormal{ker}(G)=\textnormal{core}(G)=\{1,2,17,18\}$. Hence, $\textnormal{corona}(G)\cup N(\textnormal{core}(G))=V(G)$, and $\left|\textnormal{corona}(G)\right|+\left|\textnormal{core}(G)\right|=16+4=20=2\alpha(G)+2=2\cdot 9+2.$ Moreover, $G$ contains only $2$ odd cycles.
• Figure 2: Illustration of Case 1 in the proof of \ref{['as2']}
• Figure 3: Illustration of the proof of Cases 1 and 2 in \ref{['tas2']}.
• Figure 4: Illustration of the proof of Case 3 in \ref{['tas2']}.
• Figure 5: Illustration of the proof of Cases 4 and 5 of \ref{['tas2']}.

Theorems & Definitions (46)

• Theorem 1.1: levit2025almost
• Theorem 1.2: levit2025almost
• Theorem 3.1
• Theorem 3.3: edmonds1965pathsgallai1964maximale
• Lemma 3.4
• proof
• Theorem 3.5: lovasz1972structure
• Corollary 3.6
• proof
• Lemma 3.7