A kind of conditional vertex connectivity of star graphs

A subset F ⊂ V ( G ) is called an R 2 -vertex-cut of G if G − F is disconnected and each vertex u ∈ V ( G ) − F has at least two neighbors in G − F . The cardinality of a minimum R 2 -vertex-cut of G , denoted by κ 2 ( G ) , is the R 2 -vertex-connectivity of G . In this work, we prove that κ 2 ( S n ) = 6 ( n − 3 ) for n ≥ 4 , where S n is the n -dimensional star graph.