Calculating the number of non-intersecting routes in an Euclidean graph I have an Euclidean graph: each vertex is a point on the 2D plane, so the weight of each edge is the Euclidean distance between the vertices.  I found a geometric proof that every optimal TSP solution contains no intersections.

How many non-intersecting routes could be there? Or in other words: what is the probability to guess an optimal solution to a TSP problem if we just enumerate or sample non-intersecting routes?

Edit: I want to ignore the case that D.W. mentioned. For every path that you can swap between two neighbors vertices(If we represent the path as an array of vertices so neighbors will be two vertices with consecutive indexes) without changing its non-intersecting quality, all of those paths will be considered as one.

Edit I found that this kind of removing crossings from the graph also know as 2-OPT
There are exponentially many such routes.

Think of a sequence of $n$ diamonds.  At each diamond, you can go either left or right, independently of what you do at all other diamonds.  This leads to $2^n$ paths, each of which is non-intersecting.  Now the complete graph on those vertices contains all of these paths, plus some more, so this is a lower-bound on the number of non-intersecting paths.

Therefore, you can't hope to find the optimal one in polynomial time by guessing or enumerating non-intersecting routes.

You probably could have guessed or suspected this, from the fact that Euclidean TSP is NP-complete....