New Exact and Heuristic Algorithms for Graph Automorphism Group and Graph Isomorphism

Abstract

We describe five new algorithms, named Vsep. Four of them are for the graph automorphism group and the fifth one is for finding an isomorphism between two graphs. All nonequivalent terminal nodes-discrete partitions of the search tree are stored. This is the main difference of the exact version with the known algorithms for graph automorphism group. A new strategy is used in the exact algorithm: if during its execution the computed stabilizer orbits and order get wrong values, then the algorithm continues from a new starting point losing some of the results determined so far. The new starting point is such that the results are correct. The proposed algorithms have been tested on well-known benchmark graphs and compared with three of the most competitive known tools. The results show that for some graph families the exact Vsep algorithm outperforms these algorithms, and vice-versa for some of the others. None of the tested algorithms outperform others in all cases. The heuristic versions use reduced search trees. They are almost exact and are faster than the exact one with very rare exceptions. They are applied mainly for regular graphs. The heuristic algorithms are a new choice for the user. The experiments show that the running times of Vsep algorithms have a slight dependence on vertex labeling.