On the Intersection of All Critical Sets of a Unicyclic Graph

TL;DR

This paper investigates the relationships between critical and maximum independent sets in unicyclic graphs, establishing new equalities and properties for non-Koenig-Egervary cases, expanding understanding of graph invariants.

Contribution

It proves that for non-Koenig-Egervary unicyclic graphs, the ker and core sets are equal and characterizes the sum of corona and core set sizes.

Findings

ker(G) equals core(G) for non-Koenig-Egervary unicyclic graphs

|corona(G)| + |core(G)| = 2*alpha(G) + 1 in these graphs

difference |core(G)| - |ker(G)| can be any non-negative integer in Koenig-Egervary unicyclic graphs

Abstract

A set S is independent in a graph G if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G) is the size of a maximum matching in G. If alpha(G)+mu(G)=|V|, then G=(V,E) is called a Konig-Egervary graph. The number d_{c}(G)=max{|A|-|N(A)|} is called the critical difference of G (Zhang, 1990). By core(G) (corona(G)) we denote the intersection (union, respectively) of all maximum independent sets, while by ker(G) we mean the intersection of all critical independent sets. A connected graph having only one cycle is called unicyclic. It is known that ker(G) is a subset of core(G) for every graph G, while the equality is true for bipartite graphs (Levit and Mandrescu, 2011). For Konig-Egervary unicyclic graphs, the difference |core(G)|-|ker(G)| may equal any non-negative integer. In this paper we prove that if G is a non-Konig-Egervary unicyclic graph, then: (i) ker(G)= core(G) and (ii) |corona(G)|+|core(G)|=2*alpha(G)+1. Pay attention that |corona(G)|+|core(G)|=2*alpha(G) holds for every Konig-Egervary graph.