### 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 dened 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 dened with respect to either a canonical polygonal schema or a polygonal cutset schema, which provides an interesting tradeo, 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.

Original language | English (US) |
---|---|

Pages (from-to) | 7-32 |

Number of pages | 26 |

Journal | Journal of Graph Algorithms and Applications |

Volume | 15 |

Issue number | 1 |

State | Published - 2011 |

### Fingerprint

### ASJC Scopus subject areas

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics
- Computer Science(all)
- Computer Science Applications

### Cite this

*Journal of Graph Algorithms and Applications*,

*15*(1), 7-32.

**Planar drawings of higher-genus graphs.** / Duncan, Christian A.; Goodrich, Michael T.; Kobourov, Stephen G.

Research output: Contribution to journal › Article

*Journal of Graph Algorithms and Applications*, vol. 15, no. 1, pp. 7-32.

}

TY - JOUR

T1 - Planar drawings of higher-genus graphs

AU - Duncan, Christian A.

AU - Goodrich, Michael T.

AU - Kobourov, Stephen G

PY - 2011

Y1 - 2011

N2 - 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 dened 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 dened with respect to either a canonical polygonal schema or a polygonal cutset schema, which provides an interesting tradeo, 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.

AB - 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 dened 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 dened with respect to either a canonical polygonal schema or a polygonal cutset schema, which provides an interesting tradeo, 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.

UR - http://www.scopus.com/inward/record.url?scp=79960745546&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960745546&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:79960745546

VL - 15

SP - 7

EP - 32

JO - Journal of Graph Algorithms and Applications

JF - Journal of Graph Algorithms and Applications

SN - 1526-1719

IS - 1

ER -