Computational complexity of combinatorial surfaces

We investigate the computational problems associated with combinatorial surfaces. Specifically, we present an algorithm (based on the Brahana-Dehn-Heegaard approach) for transforming the polygonal schema of a closed triangulated surface into its canonical form in <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>) time, where <italic>n</italic> is the total number of vertices, edges and faces. We also give an <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic> + <italic>gn</italic>) algorithm for constructing canonical generators of the fundamental group of a surface of genus <italic>g</italic>. This is useful in constructing homeomorphisms between combinatorial surfaces.