Let G be a 2-edge-connected simple graph on n vertices and alpha(G) be the independent number of G. Denote by G(5) the graph obtained from a K-4 by adding a new vertex and two edges joining this new vertex to two distinct vertices of the K-4. It is proved in this paper that if when alpha(G) >= 3, d(x) + d(y) + d(z) >= 3n/2 for every 3-independent set {x, y, z} of G and when alpha(G) = n for every 2-independent set {x, y} of G, then G is not Z(3)-connected if and only if G is one of the 12 specified graphs or G can be Z(3)-contracted to one of the graphs [K-3, K-4(-), K-4, G(5)], which generali...
