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Special Issue on Selected Papers from the Seventeenth International Symposium on Graph Drawing, GD 2009
DOI: 10.7155/jgaa.00215
Planar Drawings of Higher-Genus Graphs
Vol. 15, no. 1, pp. 7-32, 2011. Regular paper.
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 R2,
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.
However, unlike traditional approaches the copies of the vertices
might not be in perfect alignment but we guarantee that their order along
the boundary is still preserved.
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.
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Submitted: December 2009.
Reviewed: August 2010.
Revised: August 2010.
Accepted: August 2010.
Final: December 2010.
Published: February 2011.
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