The graph isomorphism problem: its structural complexity
The graph isomorphism problem: its structural complexity
A note on decision versus search for graph automorphism
Information and Computation
Computing From Partial Solutions
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
On the Hardness of Graph Isomorphism
SIAM Journal on Computing
Information and Computation
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It is known that a nontrivial automorphism on a given graph is computed by using any oracle that computes a pair of vertices (u,v) such that u is mapped to v by some nontrivial automorphism. In this paper, we consider a weaker oracle acting as follows. For a given graph, the oracle returns a pair (v,b) of a vertex v and a bit b@?{0,1} with the promise that if it returns (v,0), then the vertex v is fixed by some nontrivial automorphism, but if it returns (v,1), then the vertex v is moved by some nontrivial automorphism, provided that the given graph has a nontrivial automorphism. We here note that the oracle may return an arbitrary pair as its answer in case that the given graph has no nontrivial automorphism. We then show a stronger result that such an oracle is still powerful enough to compute a nontrivial automorphism. We also show that a similar result holds for RightGA, a GA-complete problem. We further investigate the computational complexity of computing a partial solution for PrefixGA which is known to be GI-complete. For this problem, we show that, when we consider any oracle similar to one mentioned above, the oracle does not help us to solve PrefixGA unless GI @?"T^p GA.