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I was trying to find all the negative cycle vertices using the Bellman–Ford algorithm using this paper solution 7.1(b) in $O(V)$ by tracing back the predecessor subgraph.It is also stated in this answer too. But then while googling I found another Stack Overflow question it is proved that finding negative cycle vertices are NP complete. So what is it actually?

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Your question is very unclear.

There is a tremendous difference between: (a) print out one example of a negative cycle, if one exists, (b) print out all negative cycles.

The former can be done in polynomial time using Bellman-Ford. The latter cannot, as there could be exponentially many cycles, so it could take exponential time. (The latter is not NP-complete, as the problem is not even in NP.)

The solution you link to is solving problem (a).

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