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

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.