I want to prove the following problem is NP-Complete:
$$\text{Given a weighted graph } G=(V,E), \text{ a vertex } v\in V \text{ and a number } K $$
The answer is YES if there exist a path of weight $K$, starting from $v$ , goes through all vertices in $V$ exactly TWICE and return to $v$.
First, we need to show the problem is in NP, but that's easy to verify in poly-time.
Now - we need to make a reduction from an NP-complete problem to our problem. I think TSP is a good idea - but I can't find a good reduction.
I thought of transforming $k$ of the TSP to $2k$ but it did not work out. I also tried to add self edges with weight $0$ but that failed too.
Any ideas?