## Planarization of Graphs Embedded on Surfaces (1995)

Venue: | in WG |

Citations: | 6 - 1 self |

### BibTeX

@INPROCEEDINGS{Djidjev95planarizationof,

author = {Hristo N. Djidjev and Shankar M. Venkatesan},

title = {Planarization of Graphs Embedded on Surfaces},

booktitle = {in WG},

year = {1995},

pages = {62--72},

publisher = {Springer}

}

### OpenURL

### Abstract

A planarizing set of a graph is a set of edges or vertices whose removal leaves a planar graph. It is shown that, if G is an n-vertex graph of maximum degree d and orientable genus g, then there exists a planarizing set of O( p dgn) edges. This result is tight within a constant factor. Similar results are obtained for planarizing vertex sets and for graphs embedded on nonorientable surfaces. Planarizing edge and vertex sets can be found in O(n + g) time, if an embedding of G on a surface of genus g is given. We also construct an approximation algorithm that finds an O( p gn log g) planarizing vertex set of G in O(n log g) time if no genus-g embedding is given as an input. 1 Introduction A graph G is planar if G can be drawn in the plane so that no two edges intersect. Planar graphs arise naturally in many applications of graph theory, e.g. in VLSI and circuit design, in network design and analysis, in computer graphics, and is one of the most intensively studied class of graphs [2...

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10921 |
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- Harary
- 1969
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- Gross, Tucker
- 1987
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- Djidjev
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- Diks, Djidjev, et al.
- 1988
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- Giblin, Graphs, et al.
- 1977
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- 1984
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Genus Reduction in Nonplanar Graphs, manuscript
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- 1974
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- Miller
- 1986
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- Sykora, Vrto
- 1993
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