A direct 3-SAT reduction proves NP-completeness of non-crossing Hamiltonian path and cycle in embedded non-planar graphs, avoiding planar crossover gadgets.
Kano.Discrete Geometry on Red and Blue Points in the Plane — A Sur- vey —, pages 551–570
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NP-Hardness of Non-Crossing Hamiltonian Path and Cycle in Non-Planar Graphs
A direct 3-SAT reduction proves NP-completeness of non-crossing Hamiltonian path and cycle in embedded non-planar graphs, avoiding planar crossover gadgets.