Let $G$ and $H$ be two connected graphs and $K$ be the disjoint union of them. Also assume that we can find generating permutation for $Aut(X)$ or any graph $X$. $G$ and $H$ are isomorphic if and only if some generator of $Aut(K)$ interchanges the two connected components. How do we prove this last statement? Please give the intuitive explanation.