A kind of conditional vertex connectivity of star graphs

A subset F⊂V(G)F⊂V(G) is called an R2R2-vertex-cut of GG if G−FG−F is disconnected and each vertex u∈V(G)−Fu∈V(G)−F has at least two neighbors in G−FG−F. The cardinality of a minimum R2R2-vertex-cut of GG, denoted by κ2(G)κ2(G), is the R2R2-vertex-connectivity of GG. In this work, we prove that κ2(Sn)=6(n−3)κ2(Sn)=6(n−3) for n≥4n≥4, where SnSn is the nn-dimensional star graph.