Early work to this end was done by Xu and Johnson, who developed multiple concepts of dissecting Murcko frameworks into constituting ring systems or abstracting them into reduced representations. They used these concepts to assign molecular equivalence numbers to molecular structures and thus classify them within chemical libraries . in their hierarchical scaffold clustering (HierS) approach use a scaffold definition similar to Murcko frameworks but additionally include all atoms that are directly attached to rings and linkers via multiple bonds. Non-cyclic molecules are taken into consideration as well and are assigned scaffolds based on their multiple bonds. To build a scaffold hierarchy, the original scaffold extracted from each molecule is dissected into its smaller parent scaffolds first. This is done by generating all smaller scaffolds that can result from the stepwise removal of ring systems, i.e. isolated single rings or fused multiple rings that share bonds or atoms, from the original scaffold. After the removal of one ring, linker atoms that have become side chains are also removed. The process is finished when only the individual ring systems are left. Via a substructure search, the scaffold hierarchy is constructed in the second step by linking parent and child scaffold if the smaller parent scaffold is a substructure of the bigger child scaffold. In the end, a tree-like hierarchy results with the individual ring systems as roots at the top, and their combinations in more complex scaffolds on the following levels. A scaffold that is not a single ring system has multiple parents in the hierarchy.