Local restricted edge connectivity and restricted edge connectivity of graphs

Let G be a connected graph and k >= 1 be an integer. Local k-restricted edge connectivity lambda(k) (X, Y) of X, Y in G is the maximum number of the edge disjoint X-Y paths for X,Y subset of V with vertical bar X vertical bar = vertical bar Y vertical bar = k,X boolean AND Y = empty set, G[X] and G[Y] are connected. k-restricted edge connectivity of G is defined as lambda(k)(C) = min{lambda(k)(X,Y) : X,Y subset of V, vertical bar X vertical bar = vertical bar Y vertical bar = k,X boolean AND Y = empty set,G[X] and G[Y] are connected}. Then G is local optimal k-restricted edge connected, if lambda(k)(X,Y) =min{omega(X),omega(Y)} for all X,Y subset of V with vertical bar X vertical bar = vertical bar Y vertical bar = k, G[X] and G[Y] are connected, where omega(X) = vertical bar[X, X]vertical bar. If lambda(k)(G) = xi(k)(G) where xi(k)(G) = min{omega(X) : U subset of V, vertical bar U vertical bar = k and G[U] is connected}, then G is called lambda(k)-optimal. In this paper, we obtain several sufficient conditions for a graph to be lambda(3)-optimal (or local optimal k-restricted edge connected).