Irredundant and perfect neighborhood sets in graphs and claw-free graphs

Abstract

Abstract Let θ ( G ), θ i ( G ), ir( G ), sir( G ) be the minimum cardinality of, respectively, a perfect neighborhood set, an independent perfect neighborhood set, a maximal irredundant set and a semi-maximal irredundant set of a graph G . It is clear that θ ( G )⩽ θ i ( G ) and that sir( G )⩽ir( G ). It has been conjectured in [5] that θ ( G )⩽ir( G ) for any graph G . In the first part of this paper we give a counter-example showing that the difference θ ( G ) − ir( G ) can be arbitrarily large. In the second part we prove that for claw-free graphs, θ ( G ) = θ i ( G )⩽sir( G ). We also describe the ( K 1,3 , B 1,3 )-free graphs for which θ ( G ) = sir( G )⩾3 and the ( K 1,3 , B 1,3 , C 6 )-free graphs for which θ ( G ) = sir( G ) = 2, where the graphs B 1,3 and C 6 are shown in Fig. 1.