Accelerations for Graph Isomorphism

Abstract

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^{3}) $. Second, without that conjecture, we prove the algorithm to be of quasi-polynomial time $ O(n^{1.5\log 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.