Planar Drawings of Higher-Genus Graphs
Abstract
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.
Keywords
Cite
@article{arxiv.0908.1608,
title = {Planar Drawings of Higher-Genus Graphs},
author = {Christian A. Duncan and Michael T. Goodrich and Stephen G. Kobourov},
journal= {arXiv preprint arXiv:0908.1608},
year = {2009}
}
Comments
A condensed version of this paper is to appear in Graph Drawing 2009. This is just a first draft. A final draft will appear in the near future