On the Complexity of Counting the Number of Vertices Moved by Graph Automorphisms

We consider the problem of deciding whether a given graph G has an automorphism which moves at least k vertices (where k is a function of ∣ V(G) ∣), a question originally posed by Lubiw (1981). Here we show that this problem is equivalent to the one of deciding whether a graph has a nontrivial automorphism, for k ∈ O(logn / loglogn ).It is commonly believed that deciding isomorphism between two graphs is strictly harder than deciding whether a graph has a nontrivial automorphism. Indeed, we show that an isomorphism oracle would improve the above result slightly–using such an oracle, one can decide whether there is an automorphism which moves at least k′ vertices, where k′ ∈ O(logn).If P $\ne$ NP and Graph Isomorphism is not NP-complete, the above results are fairly tight, since it is known that deciding if there is an automorphism which moves at least nε vertices, for any fixed ε ∈ (0, 1) , is NP-complete. In other words, a substantial improvement of our result would settle some fundamental open problems about Graph Isomorphism.