Degree sum of 3 independent vertices and -connectivity

Let G be a 2-edge-connected simple graph on n vertices and α ( 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 α ( G ) ≥ 3 , d ( x ) + d ( y ) + d ( z ) ≥ 3 n / 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 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 generalize the results of Luo et al. [R. Luo, R. Xu, J. Yin, G. Yu, Ore-condition and Z 3 -connectivity, European J. Combin. 29 (2008) 1587–1595].