Enumerating tree-like chemical graphs with given upper and lower bounds on path frequencies

Enumeration of chemical graphs satisfying given constraints is one of the fundamental problems in chemoinformatics and bioinformatics since it leads to a variety of useful applications including structure determination of novel chemical compounds and drug design.

In this paper, we consider the problem of enumerating all tree-like chemical graphs from a given set of feature vectors, which is specified by a pair of upper and lower feature vectors, where a feature vector represents the frequency of prescribed paths in a chemical compound to be constructed. This problem can be solved by applying the algorithm proposed by Ishida et al. to each single feature vector in the given set, but this method may take much computation time because in general there are many feature vectors in a given set. We propose a new exact branch-and-bound algorithm for the problem so that all the feature vectors in a given set are handled directly. Since we cannot use the bounding operation proposed by Ishida et al. due to upper and lower constraints, we introduce new bounding operations based on upper and lower feature vectors, a bond constraint, and a detachment condition.

Our proposed algorithm is useful for enumerating tree-like chemical graphs with given upper and lower bounds on path frequencies.