Almost Bipartite non-König-Egerváry Graphs Revisited

TL;DR

The paper studies almost bipartite non-König-Egerváry graphs with a unique odd cycle $C$, using invariants such as the independence number and maximum matching along with critical independent sets (diadem, nucleus, core, corona) to derive exact formulas for the vertex-robust index varrho_v(G). It establishes a partition of $V(G)$ into $V(C)$ and $N_G[diadem(G)]$, and proves key equalities: varrho_v(G) = n(G) + d(G) - xi(G) - beta(G), varrho_v(G) = corona(G) - diadem(G), and varrho_v(G) = |V(C)| + |nucleus(G)| - |core(G)|, with |V(C)| ≤ varrho_v(G) and that varrho_v(G) = |V(G)| iff G = C_{2k+1}. The work further shows that ker(G) = core(G) for these graphs and links the equality varrho_v(G) = |V(C)| to core(G) = nucleus(G), illuminating the 1-König-Egerváry boundary and the interplay among core, corona, and diadem. These results sharpen structural understanding of near-bipartite graphs and suggest directions to characterize related 1-König-Egerváry graphs and nucleus/core relations.

Abstract

Let $α(G)$ denote the cardinality of a maximum independent set, while $μ(G)$ be the size of a maximum matching in $G=\left( V,E\right) $. It is known that if $α(G)+μ(G)=\left\vert V\right\vert $, then $G$ is a König-Egerváry graph. The critical difference $d(G)$ is $\max\{d(I):I\in\mathrm{Ind}(G)\}$, where $\mathrm{Ind}(G)$\ denotes the family of all independent sets of $G$. If $A\in\mathrm{Ind}(G)$ with $d\left( X\right) =d(G)$, then $A$ is a critical independent set. For a graph $G$, let $\mathrm{diadem}(G)=\bigcup\{S:S$ is a critical independent set in $G\}$, and $\varrho_{v}\left( G\right) $ denote the number of vertices $v\in V\left( G\right) $, such that $G-v$ is a König-Egerváry graph. A graph is called almost bipartite if it has a unique odd cycle. In this paper, we show that if $G$ is an almost bipartite non-König-Egerváry graph with the unique odd cycle $C$, then the following assertions are true: 1. every maximum matching of $G$ contains $\left\lfloor {V(C)}/{2}\right\rfloor $ edges belonging to $C$; 2. $V(C)\cup N_{G}\left[ \mathrm{diadem}\left( G\right) \right] =V$ and $V(C)\cap N_{G}\left[ \mathrm{diadem}\left( G\right) \right] =\emptyset$; 3. $\varrho_{v}\left( G\right) =\left\vert \mathrm{corona}\left( G\right) \right\vert -\left\vert \mathrm{diadem}\left( G\right) \right\vert $, where $\mathrm{corona}\left( G\right) $ is the union of all maximum independent sets of $G$; 4. $\varrho_{v}\left( G\right) =\left\vert V\right\vert $ if and only if $G=C_{2k+1}$ for some integer $k\geq1$.

Figures (7)

• Figure 1: $1$-König-Egerváry graphs
• Figure 2: Almost bipartite non-König-Egerváry graphs.
• Figure 3: An almost bipartite non-König-Egerváry graph and its $D_{y}$.
• Figure 4: Both $G_{1}$ and $G_{2}$ are almost bipartite non-König-Egerváry.
• Figure 5: $1$-König-Egerváry graphs.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10