Whittlesey 12,13,14] gave a criterion which decides when two nite 2-dimensional complexes are homeomorphic. We show that graph isomorphism can be reduced eeciently to 2-complex homeomorphism, and that Whittlesey's criterion can be reduced eeciently to graph isomorphism. Therefore graph isomorphism and 2-complex homeomorphism are polynomial-time equivalent.

Canonical labeling of a graph consists of assigning a unique label to each vertex such that the labels are invariant under isomorphism. Such a labeling can be used to solve the graph isomorphism problem. We give a simple, linear time, high probability algorithm for the canonical labeling of a G(n, p) random graph for p ∈ [ ω ( ln n/n ln lnn ) , 1− ω ( ln n/n ln lnn )] . Our result covers a gap ...

The convex hull ψn,n of certain (n!) 2 tensors was considered recently in connection with graph isomorphism. We consider the convex hull ψn of the n! diagonals among these tensors. We show: 1. The polytope ψn is a face of ψn,n. 2. Deciding if a graph G has a subgraph isomorphic to H reduces to optimization over ψn. 3. Optimization over ψn reduces to optimization over ψn,n. In particular, this i...

We report the current state of the graph isomorphism problem from the practical point of view. After describing the general principles of the refinement-individualization paradigm and proving its validity, we explain how it is implemented in several of the key programs. In particular, we bring the description of the best known program nauty up to date and describe an innovative approach called ...

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 ...

We introduce a nonlocal game that captures and extends the notion of graph isomorphism. This game can be won in the classical case if and only if the two input graphs are isomorphic. Thus, by considering quantum strategies we are able to define the notion of quantum isomorphism. We also consider the case of more general non-signalling strategies, and show that such a strategy exists if and only...

The question of whether there is a polynomial time algorithm deciding whether two graphs are isomorphic has been a one of the best known open problems in theoretical computer science for more than forty years. Indeed, the graph isomorphism problem is one of the very few natural problems in NP that is neither known to be in P nor known to be NP-complete. The question is still wide open, but a nu...