Degree sum of 3 independent vertices and Z3-connectivity

Let G be a 2-edge-connected simple graph on n vertices and α(G) be the independent number of G. Denote by G5 the graph obtained from a K4 by adding a new vertex and two edges joining this new vertex to two distinct vertices of the K4. It is proved in this paper that if when α(G)≥3, d(x)+d(y)+d(z)≥3n/2 for every 3-independent set {x,y,z} of G and when α(G)≤2, d(x)+d(y)≥n for every 2-independent set {x,y} of G, then G is not Z3-connected if and only if G is one of the 12 specified graphs or G can be Z3-contracted to one of the graphs {K3,K4−,K4,G5}, which generalize the results of Luo et al. [R. Luo, R. Xu, J. Yin, G. Yu, Ore-condition and Z3-connectivity, European J. Combin. 29 (2008) 1587-1595].