Isomorphism, Automorphism Partitioning, and Canonical Labeling Can Be Solved in Polynomial-Time for Molecular Graphs

The graph isomorphism problem belongs to the class of NP problems, and has been conjectured intractable, although probably not NP-complete. However, in the context of chemistry, because molecules are a restricted class of graphs, the problem of graph isomorphism can be solved efficiently (i.e., in polynomial-time). This paper presents the theoretical results that for all molecules, the problems of isomorphism, automorphism partitioning, and canonical labeling are polynomial-time problems. Simple polynomial-time algorithms are also given for planar molecular graphs and used for automorphism partitioning of paraffins, polycyclic aromatic hydrocarbons (PAHs), fullerenes, and nanotubes.