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 $\cal S$ of genus g and produce a planar drawing of G in R2, with a bounding face defined by a polygonal schema $\cal P$ for $\cal S$. Our drawings are planar, but they allow for multiple copies of vertices and edges on $\cal P$'s boundary, which is a common way of visualizing higher-genus graphs in the plane. 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.

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