Computing Complete Graph Isomorphisms and Hamiltonian Cycles from Partial Ones

TL;DR

This paper demonstrates that finding a specific vertex mapping in graph isomorphisms is as computationally difficult as finding the entire isomorphism, and similarly for Hamiltonian cycles from partial information.

Contribution

It proves that computing a single vertex pair in an isomorphism is as hard as computing the full isomorphism, refining previous results, and extends similar hardness results to Hamiltonian cycles.

Findings

Computing a vertex pair in an isomorphism is as hard as computing the entire isomorphism.

Hardness results for deriving Hamiltonian cycles from partial cycles.

Improves upon previous complexity results by Gál et al.

Abstract

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