Before revision: We resolve in the affirmative conjectures of A. Skopenkov and Repov\v{s (1998), and M. Skopenkov (2003) generalizing the classical Hanani--Tutte  theorem to the setting of approximating maps of graphs in the plane by embeddings.
After revision: We resolve in the affirmative conjectures of A. Skopenkov and Repov\v{s (1998), and M. Skopenkov (2003) generalizing the classical Hanani--Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings.