crossing edges in the travelling salesman problem

Question

Does there exist a travelling salesman problem where the optimal solution has edges that cross?

The nodes are in an x-y plane, so crossing in this case means if you wer

Answer 1

Trivially, any connected graph where every node has two edges has only one TPS solution, and if drawn with crossings will meet your stated criteria.

If you put other constraints, such as if you were modelling travelling around the globe using trade winds, so costs are only somewhat related to position in space, you might find a more complex case where crossing is optimal.