Note: On the intersection of all critical sets of a unicyclic graph

• Authors: Vadim E. Levit;Eugen Mandrescu
• Venue: Discrete Applied Mathematics
• Year: 2014

A set S@?V is independent in a graph G=(V,E) if no two vertices from S are adjacent. The independence number@a(G) is the cardinality of a maximum independent set, while @m(G) is the cardinality of a maximum matching in G. If @a(G)+@m(G)=|V|, then G is a Konig-Egervary 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)@?A0@?}. 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 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@a(G)+1. Pay attention that |corona(G)|+|core(G)|=2@a(G) holds for every Konig-Egervary graph (Levit, 2011) [11].