Before revision:  Although we show that deciding the distances is NP-complete for general embedded graphs, we prove that our approach yields polynomial time algorithms for the distances based on the Fr\'echet distance if the graphs are trees, and for the distances based on the weak Fr\'echet distance if the graphs are trees or if they are plane.
After revision:  Although we show that deciding the distances is NP-complete for general embedded graphs, we prove that our approach yields polynomial time algorithms for the distances based on the Fr\'echet distance if the graphs are trees, and for the distance based on the weak Fr\'echet distance if the graphs are trees or if they are plane.