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

TL;DR

This paper investigates the properties of Konig-Egervary graphs, focusing on how many vertices or edges can be removed while preserving this property, and provides formulas and bounds related to these counts.

Contribution

It introduces an explicit formula for RV(G) and a tight inequality for RE(G) in Konig-Egervary graphs, enhancing understanding of their structural resilience.

Findings

Derived an equality for RV(G) based on graph parameters.

Established a tight inequality bounding RE(G).

Extended understanding of the structural properties of Konig-Egervary graphs.

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.