  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.

Group Theory