$N$ points are located in 2D plane. Some of the pair of the points are connected by line segments. What is the complexity of the problem of existence of Hamiltonian non intersecting path? What if we consider it in special cases of graphs.
Hamiltonian path problem remains NP-complete in planar graphs , so your problem is also NP-complete since in a planar graph, two edges cannot intersect with each other.
 Garey, M. R., Johnson, D. S., & Tarjan, R. E. (1976). The planar Hamiltonian circuit problem is NP-complete. SIAM Journal on Computing, 5(4), 704-714.