Problem : Let $G$ be a weighted complete graph with a non-negative weight function w. For fixed vertices $u, v \in V (G)$, where $u \neq v$ we would like to find a minimum Hamilton path from $u$ to $v$. Explain how to convert this problem to the problem of a minimum Hamilton cycle in an appropriate graph $G^∗$.
I have seen this idea for showing a ham path exists if a ham cycle exists : Modify your graph by adding another node that has edges to all the nodes in the original graph.
If the original graph has a Hamiltonian Path, the new graph will have a Hamiltonian Circuit: the circuit will run from the new node to the start node of the Path, through all the nodes along the Path, back to the new node.
But I am unsure for how to make this work for weight functions, and moreover for 2 specific vertices. Hints much appreciated.
As an aside, is it true that there will always exist a hamiltonian path with every pair of fixed starting and ending vertices in a complete graph?