On the connectivity of the non-generating graph

Given a 2-generated finite group G, the non-generating graph of G has as vertices the elements of G and two vertices are adjacent if and only if they are distinct and do not generate G. We consider the graph Sigma(G) obtained from the non-generating graph of G by deleting the universal vertices. We prove that if the derived subgroup of G is not nilpotent, then this graph is connected, with diameter at most 5. Moreover, we give a complete classification of the finite groups G such that Sigma(G) is disconnected.