## Abstract

A drawing of a graph is a monotone drawing if for every pair of vertices u and v there is a path drawn from u to v that is monotone in some direction. In this paper we investigate planar monotone drawings in the fixed embedding setting, i.e., a planar embedding of the graph is given as part of the input that must be preserved by the drawing algorithm. In this setting we prove that every planar graph on n vertices admits a planar monotone drawing with at most two bends per edge and with at most 4n−10 bends in total; such a drawing can be computed in linear time and requires polynomial area. We also show that two bends per edge are sometimes necessary on a linear number of edges of the graph. Furthermore, we investigate subclasses of planar graphs that can be realized as embedding-preserving monotone drawings with straight-line edges. In fact, we prove that biconnected embedded planar graphs and outerplane graphs always admit such drawings, and describe linear-time drawing algorithms for these two graph classes.

Original language | English (US) |
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Pages (from-to) | 233-257 |

Number of pages | 25 |

Journal | Algorithmica |

Volume | 71 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2015 |

## Keywords

- Curve complexity
- Fixed embedding
- Monotone drawings
- Planar graph drawing
- Polynomial area

## ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics