Computing Complete Graph Isomorphisms and Hamiltonian Cycles from Partial Ones

We prove that computing a single pair of vertices that are mapped onto each other by an isomorphism $ϕ$ between two isomorphic graphs is as hard as computing $ϕ$ itself. This result optimally improves upon a result of Gál et al. We establish a similar, albeit slightly weaker, result about computing complete Hamiltonian cycles of a graph from partial Hamiltonian cycles.

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