For a finite group \(G\) and a set \(\Delta\) of prime divisors of \(| G|\), \(\Gamma_\Delta(G)\) is the graph with vertex set \(\Delta\) where distinct \(p,q\in\Delta\) are joined by an edge if and only if \(G\) contains an element of order \(pq\). The authors show that \(\Gamma_\Delta(G)\) has at least two connected components if and only if \(G\) has a nontrivial 2-connected sharp \(\Delta\)-character, i.e. if there exists a generalized character \(\chi\) of \(G\) such that \(\{\chi(g):1\neq g\in G_{\Delta'}\}=\{a,b\}\subseteq\mathbb{Z}\setminus\{\chi(1)-1,\chi(1)+1\}\), \((\chi(1)-a,\chi(1)-b)=1\) and \(| G|_{\Delta'}=(\chi(1)-a)(\chi(1)-b)\).