I've encountered a description of a well-known, hard to compute problem. Let us consider $n$ sites that need to be connected in the shortest possible way. If I am only allowed to connect the sites I have a traveling salesman problem. Now, I am allowed to add nodes to the map and connect the sites to those nodes (or even connect two nodes). What it the name of this problem?
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Assuming the sites are given as points in the plane, this is the Steiner tree problem.
By the way, if you are trying to do the same thing without introducing additional nodes, as in the initial part of your question, then this is the problem of computing a minimum spanning tree, not the travelling salesman problem, and can be solved in polynomial time.