The virtually generating graph of a profinite group

Abstract: We consider the graph $\Gamma_{\rm{virt}}(G)$ whose vertices are the elements of a finitely generated profinite group $G$ and where two vertices $x$ and $y$ are adjacent if and only if they topologically generate an open subgroup of $G$. We investigate the connectivity of the graph $\Delta_{\rm{virt}}(G)$ obtained from $\Gamma_{\rm{virt}}(G)$ by removing its isolated vertices. In particular we prove that for every positive integer $t$, there exists a finitely generated prosoluble group $G$ with the property that $\Delta_{\rm{virt}}(G)$ has precisely $t$ connected components. Moreover we study the graph $\tilde \Gamma_{\rm{virt}}(G)$, whose vertices are again the elements of $G$ and where two vertices are adjacent if and only if there exists a minimal generating set of $G$ containing them. In this case we prove that the subgraph $\tilde \Delta_{\rm{virt}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.