Fast Isomorphism Testing of Graphs with Regularly-Connected Components

The Graph Isomorphism problem has both theoretical and practical interest. In this paper we present an algorithm, called conauto-1.2, that efficiently tests whether two graphs are isomorphic, and finds an isomorphism if they are. This algorithm is an improved version of the algorithm conauto, which has been shown to be very fast for random graphs and several families of hard graphs. In this paper we establish a new theorem that allows, at very low cost, the easy discovery of many automorphisms. This result is especially suited for graphs with regularly connected components, and can be applied in any isomorphism testing and canonical labeling algorithm to drastically improve its performance. In particular, algorithm conauto-1.2 is obtained by the application of this result to conauto. The resulting algorithm preserves all the nice features of conauto, but drastically improves the testing of graphs with regularly connected components. We run extensive experiments, which show that the most popular algorithms (namely, nauty and bliss) can not compete with conauto-1.2 for these graph families.