J. Chem. I f . Comput. Sci. 1992, 32, 167-173
167
Toward the Solution of the Isomorphism Problem in Generation of Chemical Graphs: Generation of Benzenoid Hydrocarbonst IVAN P. BANGOV Bulgarian Academy of Sciences, Institute of Organic Chemistry, Building 9, Sofia 1113, Bulgaria Received August 27, 1991 A novel approach to the treatment of the isomorphism problem in generation of chemical graphs (structures) is discussed. It is based on the presumption that the isomorphic structures are the result of permutations of equivalent substructure units: atoms, fragments, etc. To avoid such permutations, a hierarchical generation scheme has been developed. One of the aims of this paper is to show how this approach can be reduced to the particular case of generation of benzenoid
compounds. INTRODUCTION
These approaches have two substantial drawbacks: (1) At each step of the generation process all possible Two graphs I? and I?* are considered isomorphic if the permutations of equivalent vertices (in ref 4 they are called following relation holds2 segments) must be generated, and the permutation producing the greatest characteristic vector or stack is selected. This leads A* = gAg-l (1) to an enormous number of permutations for structures having Here A is the adjacency matrix and g a similarity operation more than 10 atoms, Le., for most of the real-world problems. leading to simultaneous permutations between two rows and (2) There is no efficient method for handling structural two columns of.'l fragments (substructures) within this approach. The fragments For example, from the two adjacency matrices of the isoare usually perceived by using substructure search procedures morphic graphs a and b in Figure 1, the matrix b is obtained after each structure generation6 This burdens the generation from matrix 1 by permuting both rows 1 and 2 and columns process with additional computational work. 1 and 2. we have introduced a new approach In a series of However, keeping track of all such permutations during the whose purpose is to circumvent the drawbacks just mentioned generation process is practically an impossible task. Therefore, above. a different approach has been exploited in the l i t e r a t ~ r e . ~ - ~ In this paper we summarize the general principals of our One out of all the structures belonging to an isomorphism class approach. Further, the generation of benzenoid structures is is selected as canonical on the basis of a given canonicity discussed. This new development illustrates the flexibility of predicate. Further, at each step of the structure growth the the approach which allows a reduction of the general scheme structure examines itself against this predicate by checking to some particular classes of structures. all generated permutations and only those producing canonical FUNDAMENTALS OF THE METHOD continuations are allowed. Kudo and Sasaki3s4introduced the connectivity stack with The isomorphism problem is usually related to the problem the following relationship between the adjacency matrix eleof comparing two graphs in order to establish whether they ments aij and their position in the stack k: are identical. It emerges from the necessity that the graph vertices be labeled by numbering them. For a structure of n k = i 0'- 1)U - 2)/2 (i
