Algorithm detect simple curves using Voronoi diagram or Delaunay triangulation?

Question

I wonder if there is algorithm/method to determine if closed (or even non closed) curve is simple or not, using the mathematics from the field of computational geometry? Especially I wonder if exist solution using Voronoi diagrams or Delaunay triangulation?

I'm not the strongest math man but from some reason I thought about it and it seems possible.

Example demonstrating the issue is here.

Thanks,