Ladner [18] proved the existence of an in nite hierarchy of problems of interme-diate complexity assuming that P is di erent from NP. The graph isomorphism problem, for reasons stated above, is believed to be a natural example. In this article, we study graph isomorphism and related problems. There is now a vast literature on graph isomorphism On parallel complexity of the subgraph homeomorphism and the subgraph isomorphism problem for classes of planar graphs We also show that the related subgraph isomorphism problem for two-connected outerplanar graphs is in NC 3. This is the first example of a restriction of subgraph isomorphism to a non-trivial graph family admitting an NC Graph isomorphism is not known to be in BQP. There has been a lot of work done on trying to put it in. A very intriguing observation is that graph isomorphism could be solved if quantum computers could solve the non-abelian hidden subgroup problem for the symmetric group (factoring and discrete log are solved by using the abelian hidden subgroup problem, which in turn is solved by applying the We show that the Graph Isomorphism (GI) problem and the related problems of String Isomorphism(under groupaction) (SI) and Coset Intersection (CI) can be solved in quasipolynomial(exp (logn)O(1)) time. The best previousbound forGI wasexp(O(√ nlogn)), where nis the number of vertices (Luks, ); for the other two problems, the bound was

$\begingroup$ Please check complexity class GA and the book cited there. (The definition of the graph automorphism problem in the Complexity Zoo is technically incorrect; this problem is usually defined as the decision problem to decide whether a given graph has any nontrivial automorphism.) $\endgroup$ – Tsuyoshi Ito Sep 12 '12 at equivalent to GI (so-called isomorphism-complete problems), problems to which GI is Turing reducible (like Group Factorization) and problems that seem to be incomparable with GI (like Group Intersection). The present work makes a contribution towards a better (structural) complexity classiﬁcation of Graph Isomporhism and other related The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic.. Besides its practical importance, the graph isomorphism problem is a curiosity in computational complexity theory as it is one of a very small number of problems belonging to NP neither known to be solvable in polynomial time nor NP-complete: it is one of only 12 such problems +complexity/en-en. The graph isomorphism problem: its structural complexity. Boston, MA: Birkhäuser. Google Scholar; Michael Koren (). Pairs of sequences with a unique realization by bipartite graphs. Journal of Combinatorial Theory, Series B 21(3), Google Scholar Cross Ref; Andreas Krebs & Oleg Verbitsky ().