On Konig-Egervary Collections of Maximum Critical Independent Sets

TL;DR

This paper proves that if all maximum critical independent sets in a graph form a Konig-Egervary collection, then the graph itself is a Konig-Egervary graph, extending previous conjectures in graph theory.

Contribution

It establishes a new characterization of Konig-Egervary graphs based on the properties of maximum critical independent sets.

Findings

If the family of all maximum critical independent sets forms a Konig-Egervary collection, then the graph is a Konig-Egervary graph.

Generalizes a recent conjecture about the structure of such graphs.

Provides a new perspective on the relationship between critical independent sets and graph classes.

Abstract

Let G be a simple graph with vertex set V(G). A set S is independent if no two vertices from S are adjacent. The graph G is known to be a Konig-Egervary if alpha(G)+mu(G)= |V(G)|, where alpha(G) denotes the size of a maximum independent set and mu(G) is the cardinality of a maximum matching. The number d(X)= |X|-|N(X)| is the difference of X, and an independent set A is critical if d(A) = max{d(I):I is an independent set in G} (Zhang; 1990). Let Omega(G) denote the family of all maximum independent sets. Let us say that a family Gamma of independent sets is a Konig-Egervary collection if |Union of Gamma| + |Intersection of Gamma| = 2alpha(G) (Jarden, Levit, Mandrescu; 2015). In this paper, we show that if the family of all maximum critical independent sets of a graph G is a Konig-Egervary collection, then G is a Konig-Egervary graph. It generalizes one of our conjectures recently validated in (Short; 2015).