Subgraph Isomorphism in Polynomial Time

In this paper, we propose a new approach to the problem of subgraph isomorphism detection. The new method is designed for systems which di erentiate between graphs that are a priori known, so-called model graphs, and unknown graphs, so-called input graphs. The problem to be solved is to nd a subgraph isomorphism from an input graph, which is given on-line, to any of the model graphs. The new method is based on an intensive preprocessing step in which the model graphs are used to create a decision tree. At run time, the input graph is then classi ed by the decision tree and all model graphs for which there exists a subgraph isomorphism from the input graph are detected. If we neglect the time needed for preprocessing, the computational complexity of the new subgraph isomorphism algorithm is only quadratic in the number of input graph vertices. Furthermore, it is independent of the number of model graphs and the number of edges in any of the graphs. However, the decision tree that is constructed in the preprocessing step may grow exponentially with the number of vertices of the model graphs. Therefore, we present several pruning techniques which aim at reducing the size of the decision tree. A computational complexity analysis of the new method is given. Also, the advantages and disadvantages of the new algorithm are studied in a number of practical experiments with randomly generated graphs. Finally, the application of the algorithm in a graphic symbol recognition system is brie y discussed.