  Computational topology is an area that revisits topological problems from an
algorithmic point of view, and develops topological tools for improved
algorithms. We survey results in computational topology that are concerned with
graphs drawn on surfaces. Typical questions include representing surfaces and
graphs embedded on them computationally, deciding whether a graph embeds on a
surface, solving computational problems related to homotopy, optimizing curves
and graphs on surfaces, and solving standard graph algorithm problems more
efficiently in the case of surface-embedded graphs.
