On the Number of Vertices/Edges whose Deletion Preserves the Konig-Egervary Property

TL;DR

This work investigates how deleting vertices or edges preserves the König-Egerváry property, focusing on graphs where $\alpha(G)+\mu(G)=n(G)$. By leveraging critical independent sets and the associated core/ker structure, the authors derive a precise vertex heredity formula $\varrho_v(G)=n(G)-\xi(G)+\varepsilon(G)$ for König-Egerváry graphs, and establish a tight, generally stronger-than-baseline edge heredity bound $\varrho_e(G)\ge m(G)-\xi(G)+\varepsilon(G)$. The analysis hinges on the behavior of induced subgraphs on critical sets, the decomposition $G=G[X]\cup G-X$ with $X=A\cup N(A)$, and the extendability of maximum matchings, yielding deep links between deletions and maximum matching structure. These results advance understanding of how near-König-Egerváry properties behave under vertex/edge deletions and connect to Lorentzen-type inequalities via the critical-set framework.

Abstract

The graph G=(V,E) is called Konig-Egervary if the sum of its independence number and its matching number equals its order. Let RV(G) denote the number of vertices v such that G-v is Konig-Egervary, and let RE(G) denote the number of edges e such that G-e is Konig-Egervary. Clearly, RV(G) = |V| and RE(G) = |E| for bipartite graphs. Unlike the bipartiteness, the property of being a Konig-Egervary graph is not hereditary. In this paper, we present an equality expressing RV(G) in terms of some graph parameters, and a tight inequality bounding RE(G) in terms of the same parameters, when G is Konig-Egervary.

Figures (12)

• Figure 1: $G_{1}-v_{1}$, $G_{2}-e_{2}$ are König-Egerváry graphs, while $G_{1}-v_{2}$ and $G_{2}-e_{1}$ are not König-Egerváry graphs.
• Figure 2: $S\in\Omega(G)$ and $A$ is a critical independent set of $G$.
• Figure 3: $G$ is a non-König-Egerváry graph with core$(G)=\{a,b,c,u\}$.
• Figure 4: $d\left( G_{1}\right) =d\left( G_{2}\right) =1$, while only $G_{1}$ is not a König-Egerváry graph.
• Figure 5: $G_{1}$, $G_{2}$ and $G_{3}$ are König-Egerváry graphs

Theorems & Definitions (46)

• Lemma 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Proposition 1.8
• Corollary 1.9
• Theorem 2.1