On the Intersection of All Critical Sets of a Unicyclic Graph

A set S ⊆ V is independent in a graph G = (V, E) if no two vertices from S are adjacent. The independence number α(G) is the cardinality of a maximum independent set, while μ(G) is the size of a maximum matching in G. If α(G) + μ(G) = |V |, then G is called a König-Egerváry graph. The number dc (G) = max{|A| − |N (A)| : A ⊆ V } is called the critical difference of G [21], where N (A) = {v : v ∈ V,N (v) ∩A 6= ∅}. 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 the relation ker (G) ⊆ core(G) holds for every graph G [13], while the equality is true for bipartite graphs [14]. 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 [14].