Monotonic Properties of Collections of Maximum Independent Sets of a Graph

TL;DR

This paper investigates the properties of maximum independent sets in graphs, introducing a new function that exhibits monotonic behavior and characterizing collections related to Konig-Egervary graphs.

Contribution

It defines a new monotonic function on collections of maximum independent sets and characterizes Konig-Egervary collections, extending understanding of their structural properties.

Findings

The function f is '<<'-increasing with respect to certain set orderings.

Families called Konig-Egervary collections have specific size properties.

Subcollections of Konig-Egervary collections are also Konig-Egervary.

Abstract

Let G be a simple graph with vertex set V(G). A subset S of V(G) 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. Let Omega(G) denote the family of all maximum independent sets, and f be the function from the set of subcollections Gamma of Omega(G) such that f(Gamma) = (the cardinality of the union of elements of Gamma) + (the cardinality of the intersection of elements of Gamma). Our main finding claims that f is "<<"-increasing, where the preorder {Gamma1} << {Gamma2} means that the union of all elements of {Gamma1} is a subset of the union of all elements of {Gamma2}, while the intersection of all elements of {Gamma2} is a subset of the intersection of all elements of {Gamma1}. Let us say that a family {Gamma} is a Konig-Egervary collection if f(Gamma) = 2*alpha(G). We conclude with the observation that for every graph G each subcollection of a Konig-Egervary collection is Konig-Egervary as well.