On the intersection of all critical sets of a unicyclic graph

Abstract

A set S⊆V is independent in a graph G=(V,E) if no two vertices from S are adjacent. The independence numberα(G) is the cardinality of a maximum independent set, while μ(G) is the cardinality of a maximum matching in G. If α(G)+μ(G)=|V|, then G is a König–Egerváry graph. The number d(G)=max{|A|−|N(A)|:A⊆V} is the critical difference of G (Zhang, 1990) [22], where N(A)={v:v∈V,N(v)∩A≠0̸}.By core(G) (corona(G)) we denote the intersection (union, respectively) of all maximum independent sets, and by ker(G) we mean the intersection of all critical sets. A connected graph having only one cycle is called unicyclic.It is known that the relation ker(G)⊆ core (G) holds for every graph G (Levit, 2012) [14], while the equality is true for bipartite graphs (Levit, 2013) [15]. For König–Egerváry 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-König–Egerváry unicyclic graph, then: (i) ker(G)=core(G) and (ii) |corona(G)|+|core(G)|=2α(G)+1. Pay attention that |corona(G)|+|core(G)|=2α(G) holds for every König–Egerváry graph (Levit, 2011) [11].