In the original graph representation, new operator nodes were created for all appearance of an operator. For example, take a SAT problem. If in a SAT problem there were 4 clauses containing $\neg x_2$, then there would be 4 appearances of the operator $\neg$. Instead, a more compact version would only require one $\neg$ operator and all clauses using $\neg x_2$ would be connected to this one. This achieves the primary aims of the original reduction that was done: reducing the number of nodes and explicitly representing the connection between variable nodes and their negated versions.
In the original graph representation, new operator nodes were created for all appearance of an operator. For example, take a SAT problem. If in a SAT problem there were 4 clauses containing $\neg x_2$, then there would be 4 appearances of the operator $\neg$. Instead, a more compact version would only require one $\neg$ operator and all clauses using $\neg x_2$ would be connected to this one. This achieves the primary aims of the original reduction that was done: reducing the number of nodes and explicitly representing the connection between variable nodes and their negated versions.