  In this paper, we give polynomial-time algorithms that can take a graph G
with a given combinatorial embedding on an orientable surface S of genus g and
produce a planar drawing of G in R^2, with a bounding face defined by a
polygonal schema P for S. Our drawings are planar, but they allow for multiple
copies of vertices and edges on P's boundary, which is a common way of
visualizing higher-genus graphs in the plane. Our drawings can be defined with
respect to either a canonical polygonal schema or a polygonal cutset schema,
which provides an interesting tradeoff, since canonical schemas have fewer
sides, and have a nice topological structure, but they can have many more
repeated vertices and edges than general polygonal cutsets. As a side note, we
show that it is NP-complete to determine whether a given graph embedded in a
genus-g surface has a set of 2g fundamental cycles with vertex-disjoint
interiors, which would be desirable from a graph-drawing perspective.
