An Effective Topological Symmetry Perception and Unique Numbering Algorithm

Determination of equivalence classes of atoms in molecules and the unique numbering for the molecular graphs are of major interest for many structure processing tasks and many programs have been reported for this purpose. Most of them were based on the use of graph invariants, but such methods reportedly failed to give correct partitioning for certain structures and the only theoretically rigorous method is based on atom-by-atom matchings which was considered to be computationally impractical. In order to avoid the failures of partitioning and the time-consuming atom-by-atom matching, on the basis of a profound analysis on the mechanism of Morgan algorithm, this work proposed two improvements for the original Morgan algorithm. The first improvement is to avoid the oscillatory behavior of Morgan algorithm. The second improvement referred to as single-vertex Morgan algorithm, is to decompose the Morgan algorithm into single-vertex processing. By incorporating these improvements, an effective topological symmetry perception and unique numbering algorithms were devised. The high performance of these algorithms is demonstrated with some graphs that are difficult to partition.