On Restricted Connectivity and Extra Connectivity of Hypercubes and Folded Hypercubes

Given a graph G and a non-negative integer h, the h-restricted connectivity κh (G) of G is the minimum cardinality of a set of vertices of G, in which at least h neighbors of any vertex is not included, if any, whose deletion disconnects G and every remaining component has the minimum degree of vertex at least h;and the h-extra connectivity κh(G) of G is the minimum cardinality of a set of vertices of G, if any, whose deletion disconnects G and every remaining component has order more than h. This paper shows that for the hypercube Qn and the folded hypercube FQn, κ1(Qn)=κ(1)(Qn)=2n-2 for n≥3, κ2(Qn)=3n-5 for n≥4, κ1(FQn)=κ(1)(FQn)=2n for n≥4andκ(2) (FQn) =4n-4 for n≥8.