Summary: Given a graph \(G\), a subset \(M\) of \(V(G)\) is a module of \(G\) if for each \(v\in V(G)\setminus M\) and \(x,y\in M, vx \in E(G)\) if and only if \(vy \in E(G)\). The trivial modules of \(G\) are \(\emptyset\), \(\{u\}\) \((u\in V(G))\) and \(V(G)\). Let \(G\) be a graph with at least four vertices. The graph \(G\) is prime if all its modules are trivial. We are interested in the minimum number \(\delta (G)\) of edge modifications (edge deletions and/or additions) that are needed to transform \(G\) into a prime graph. We prove that \(\delta (G) \le v(G)-1\), where \(v(G)=|V(G)|\), and that the upper bound is uniquely reached by the complete graphs and their complements. We then characterize the graphs \(G\) such that \(\delta (G)=v(G)-2\) or \(v(G)-3\), respectively. Lastly, we look for the \(\delta\)-critical graphs, i.e. the graphs \(G\) for which \(\delta (H)>\delta (G)\) for every graph \(H\) obtained from \(G\) by deleting one vertex. We prove that the \(\delta\)-critical graphs are precisely the half graphs and their complements.