The First Time KE is Broken up

TL;DR

This paper investigates hereditary KE collections, characterizes their properties, and explores the differences in intersections and unions within partitions, extending understanding of KE graphs and relevant collections.

Contribution

It provides new characterizations of hereditary KE collections and KE graphs, and analyzes the differences in intersections and unions in partitions of relevant collections.

Findings

New characterizations for hke collections

Insights into KE graph structures

Relationships between intersections and unions in partitions

Abstract

A relevant collection is a collection, $F$, of sets, such that each set in $F$ has the same cardinality, $\alpha(F)$. A Konig Egervary (KE) collection is a relevant collection $F$, that satisfies $|\bigcup F|+|\bigcap F|=2\alpha(F)$. An hke (hereditary KE) collection is a relevant collection such that all of his non-empty subsets are KE collections. In \cite{jlm} and \cite{dam}, Jarden, Levit and Mandrescu presented results concerning graphs, that give the motivation for the study of hke collections. In \cite{hke}, Jarden characterize hke collections. Let $\Gamma$ be a relevant collection such that $\Gamma-\{S\}$ is an hke collection, for every $S \in \Gamma$. We study the difference between $|\bigcap \Gamma_1-\bigcup \Gamma_2|$ and $|\bigcap \Gamma_2-\bigcup \Gamma_1|$, where $\{\Gamma_1,\Gamma_2\}$ is a partition of $\Gamma$. We get new characterizations for an hke collection and for a KE graph.