## Abstract

In a convex grid drawing of a plane graph, every edge is drawn as a straight-line segment without any edge-intersection, every vertex is located at a grid point, and every facial cycle is drawn as a convex polygon. A plane graph G has a convex drawing if and only if G is internally triconnected. It has been known that an internally triconnected plane graph G of n vertices has a convex grid drawing on a grid of O(n^{3}) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n^{3}) to O(n^{2}), which is optimal up to a constant factor. More precisely, we show that G has a convex grid drawing on a 2n × 4n grid. We also present an algorithm to find such a drawing in linear time.

Original language | English |
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Pages (from-to) | 347-362 |

Number of pages | 16 |

Journal | Discrete Mathematics, Algorithms and Applications |

Volume | 2 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 Sep 1 |

## Keywords

- Convex drawing
- plane graph
- triconnected component decomposition

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics