The Restricted Connectivity Of Enhanced Hypercube And Embedded Connectivity Of (n,k)-star Graphs

Let G =(V,E)be a connected graph and h be a nonnegative integer.A subset F(?)V(G)(resp.F(?)E(G))of G,if any,is called an h-restricted vertex cut(resp.h-restricted edge cut)of G,if G-F is disconnected and every vertex in G-F has at least h neighbors(?(G—F)?2).The cardinality of a minimum h-restricted vertex cut(resp.h-restricted edge cut)of G is the h-restricted connectivity(resp.h-restricted edge connectivity)of G,and denoted by Kh(G)(resp.?h(G)).For an n-dimensional recursive network G,an h-embedded vertex cut(resp.h-embedded edge cut)of G is a set of vertices(resp.edges),if any,whose deletion results in a disconnected graph and each vertex of the remaining component lies in an undam-aged h-dimensional subnetwork.The cardinality of a minimum h-embedded vertex cut(reap.h-embedded edge cut),if any,is the h-embedded vertex connectivity(resp.h-embedded edge connectivity)of G and is denoted by?h(G)(resp.?h(G)).In Chapter 2,we consider the h-restricted connectivity of Qn.k for 2?k?n-1.In Chapter 3,we investigate the h-embedded(edge)connectivity of Sn,k.Our main results are as follows.(1)kh(Qn,k)= 2h(n-h+ 1)for4?k?n-1 and 0?h?n-3,?h(Qn,k)= 2h(n-h+ 1)for 2?k?n-1 and 0?h?n-2.(2)kh(Qn,3)= 2h-1(n-h+ 1)for n?5 and 4?h?n-1,kh(Qn,2)= 2h-1(n-h+ 1)for n?4 and 3?h?n-1.(3)K3(Qn,3)= 6n-16 for n?5.(4)K2(Qn,3)= 4n-8 for n?4 and K2(Qn,2)= 3n-5 for n?3.(5)k1(Qn,3)= 2n and k1(Qn,2)= 2n-2 for n?3.(6)?h(Sn,2)=?h(Sn,2)=n-1 for 1?h?n-2.(7)?2(Sn,k)= n+k-3 for 2?k?n-1.(8)?2(Sn,k)=2n-4 for n?4 and k?3.(9)(a)?h(Sn,3)=(?)...