Semiregularity and connectivity of the non-$\mathfrak{F}$ graph of a finite group

Given a class $\mathfrak{F}$ of finite groups, we consider the graph ${\tilde{\mathrm{\Gamma }}}_{\mathfrak{F}}(G)$ whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $\langle g,h\rangle\notin\mathfrak F$\mathrm{.}Moreoverwedenoteby$$\mathcal{I}${\mathfrak F}(G)$thesetoftheisolatedverticesof$\widetilde\Gamma{\mathfrak F}(G).$Weaddressthefollowingquestion:towhatextentthefactthat$$\mathcal{I}$_{\mathfrak F}(G)$isasubgroupof$H$forany$H\leq G,$ implies that the graph ${\mathrm{\Gamma }}_{\mathfrak{F}}G)$ obtained from ${\tilde{\mathrm{\Gamma }}}_{\mathfrak{F}}(G)$ by deleting the isolated vertices is a connected graph?