On the Structure of the Minimum Critical Independent Set of a Graph

TL;DR

This paper investigates the structure of the minimum critical independent set in a graph, revealing that it equals the union of all inclusion minimal independent sets with positive difference, thus deepening understanding of graph independence properties.

Contribution

It establishes that ker(G) equals the union of all inclusion minimal independent sets with positive difference, providing new structural insights into critical independent sets.

Findings

ker(G) equals the union of minimal independent sets with positive difference

ker(G) is a subset of core(G) in all graphs

Equality of ker(G) and core(G) holds for bipartite graphs

Abstract

Let G=(V,E). A set S is independent if no two vertices from S are adjacent. 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}. Let us recall that ker(G) is the intersection of all critical independent sets, and core(G) is the intersection of all maximum independent sets. Recently, it was established that ker(G) is a subset of core(G) is true for every graph, while the corresponding equality holds for bipartite graphs. In this paper we present various structural properties of ker(G). The main finding claims that ker(G) is equal to the union of all inclusion minimal independent sets with positive difference.