Accelerations for Graph Isomorphism

In this paper, we present two main results. First, by only one conjecture (Conjecture 2.9) for recognizing a vertex symmetric graph, which is the hardest task for our problem, we construct an algorithm for finding an isomorphism between two graphs in polynomial time O(n). Second, without that conjecture, we prove the algorithm to be of quasi-polynomial time O(n ). The conjectures in this paper are correct for all graphs of size no larger than 5 and all graphs we have encountered. At least the conjecture for determining if a graph is vertex symmetric is quite true intuitively. We are not able to prove them by hand, so we have planned to find possible counterexamples by a computer. We also introduce new concepts like collapse pattern and collapse tomography, which play important roles in our algorithms.