On the core 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. By core( G ) we mean the intersection of all maximum independent sets. 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 . A connected graph having only one cycle, say C , is a unicyclic graph . In this paper we prove that if G is a unicyclic graph of order n and n − 1 = α ( G ) + μ ( G ), then core( G ) coincides with the union of cores of all trees in G − C .