Edge Fault Tolerance Of Graphs With Respect To ?₂-optimal Property

An edge cut F of a connected graph G is said to be an h-restricted edge cut if each component of G—F has at least h vertices.The h-restricted edge connectivity?h(G)of G is the minimum size over all h-restricted edge cuts.A graph G is said to be ?h-connected if ?h(G)exists.Let ?h(G)?min{|?(A)|:G|A|is connected and|A|= h?.where ?(A)is the subset of edges having exactly one end node in A and G[A]is the subgraph induced by the node set A.A ?h-connected graph G is said to be ?h-optimal if ?h(G)=?h(G).A ?h-optimal graph G is said to be m-?h-optimal if G?F is still ?h-optimal for any edge subset F(?)E(G)with |F|?m.The edge fault tolerance of a ?h-optimal graph G with respect to the ?h-optimal property,.denoted by ?h(G),is the maximum integer m such that G is m-?h-optimal.In this paper,we give the lower bound of ?2(G)for general A2-optimal graphs.In particular,we give strict lower and upper bounds of ?2(G)for k-regular ?2-optimal graphs.Besides,as applications,we determine exact values of ?2(G)for two families of networks.