How could I tell from the reported tour weight whether a Hamilton circuit of G has been found?

Question

Here is the idea of using a travelling salesman algorithm to search for a Hamilton circuit in any simple graph.

Consider this graph $G$:

TSP algorithms usually work on a complete weighted graph. One way to transform G into such a graph $G^*$ is:

Suppose the Nearest Neighbour algorithm were applied to $G^*$ . How could I tell from the reported tour weight whether a Hamilton circuit of $G$ had been found?

Answer 1

All the edges of the original graph $G$ will have weight 1 in the new graph $G^*$. All the other edges in $G^*$ have higher weights.

Suppose the Nearest Neighbour algorithm had found a valid Hamiltonian tour, that is to say, all the edges it chose are edges of the original graph. How many edges are in that tour? So what is its total weight?

Suppose the Nearest Neighbour algorithm had found an invalid Hamiltonian tour, that is to say, the Hamiltonian tour of $G^*$ it found uses some edges that are not edges of the original graph. What can you say about its total weight?