Hamiltonian path after removing a node

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We have an undirected connected graph $G$ with a hamiltonian path between nodes $a$ and $b$, if we remove a node that this removing doesn't blemish the connectivity ($G$ is still connected and the removing node is not $a$ or $b$), what conditions has to exist for this graph or the removing node that makes it a graph with no hamiltonian path between $a$ and $b$?