Planar Drawings of Higher-Genus Graphs

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.