We consider the graph $\Gamma _{\text {virt}}(G)$ 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 $\Delta _{\text {virt}}(G)$ obtained from $\Gamma _{\text {virt}}(G)$ 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 $\Delta _{\operatorname {\mathrm {virt}}}(G)$ has precisely t connected components. Moreover, we study the graph $\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$ , 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 $\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.

我们考虑图 $\Gamma _{\text {virt}}(G)$， 其顶点是有限生成的超限群G的元素，其中两个顶点x和y相邻当且仅当它们拓扑生成开子群的G。我们通过删除其孤立顶点来研究从 $\Gamma _{\text {virt}}(G)$ 获得 的图 $\Delta _{\text {virt}}(G)$ 的连通性 。特别地，我们证明对于每个正整数t，存在一个有限生成的易解群G，其性质 $\Delta _{\operatorname {\mathrm {virt}}}(G)$ 恰好具有 t连接组件。此外，我们研究了图 $\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$ ，其顶点再次是G的元素，并且当且仅当存在最小生成包含它们的G集。在这种情况下，我们证明去除孤立顶点获得的子图 $\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$ 是连通的，并且直径最大为 3。