Question: Suppose you have $4$ cities $1, 2, 3$ and $4$. You want to find all ways to go from $1$ to $4$ such as $1=4$, $1=2-2=4$, $1=3-3=4$, $1=3-3=2-2=4$ and $1=2-2=3-3=4$. Distances between cities do not matter. What is the name of this TSP variant where I need to only find the possible routes (not requiring to connect all vertices but need to connect starting-point and end-point)?

Answer: 
You are looking for the *set of paths between two vertices of a simple, unweighted graph*. This has little to do with the TSP and I don't think there is a name for it, any preciser than I stated it. 

Some kind of *depth-first search* might solve this, you may want to look on [this question on Mathoverflow](https://mathoverflow.net/questions/18603/finding-all-paths-on-undirected-graph), where the matter already has been discussed.
*I want to draw more attention to this finding...*

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> *You want to find all ways to go from 1 to 4 such as 1=4, 1=2−2=4, 1=3−3=4, 1=3−3=2−2=4 and 1=2−2=3−3=4.*
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There is an exact solution for it, check up the integer sequence [A000522](http://oeis.org/A000522) for the total number of arrangements of a set with $n$ elements.