On the

A g-extra cut of a non-complete graph G , g ≥ 0, is a set of vertices in G whose removal disconnects the graph, while every component in the survival graph contains at least g + 1 vertices. The g-extra connectivity of G then refers to the size of a minimum g -extra cut of G . The augmented hypercube, denoted by A Q n , n ≥ 3, is a rich variant of the hypercube structure. In this paper, we present a sequence of construction based upper bounds of its g -extra connectivity, study its lower bound via the super connectedness property, and suggest an asymptotically tight bound. • A result on the exact number of the common neighbors of any pair of its vertices, adjacent or not, has been derived for the augmented cube structure. • Three general processes of deriving the upper bound of the g -extra connectivity of the augmented cube are explored, leading to tighter upper bounds. • The lower bound of the above g -extra connectivity is also studied via the super connectedness property of a network structure. • These results agree with existing ones when g ∈ [ 1 , 3 ], and the result for g = 4 is apparently new. For the general case, we give an asymptotic result.