### 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, and we show that biconnected embedded planar graphs and outerplane graphs always admit such drawings, which can be computed in linear time.

Original language | English (US) |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 379-390 |

Number of pages | 12 |

Volume | 7034 LNCS |

DOIs | |

State | Published - 2012 |

Event | 19th International Symposium on Graph Drawing, GD 2011 - Eindhoven, Netherlands Duration: Sep 21 2011 → Sep 23 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7034 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 19th International Symposium on Graph Drawing, GD 2011 |
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Country | Netherlands |

City | Eindhoven |

Period | 9/21/11 → 9/23/11 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 7034 LNCS, pp. 379-390). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7034 LNCS). https://doi.org/10.1007/978-3-642-25878-7_36

**Monotone drawings of graphs with fixed embedding.** / Angelini, Patrizio; Didimo, Walter; Kobourov, Stephen G; McHedlidze, Tamara; Roselli, Vincenzo; Symvonis, Antonios; Wismath, Stephen.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 7034 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7034 LNCS, pp. 379-390, 19th International Symposium on Graph Drawing, GD 2011, Eindhoven, Netherlands, 9/21/11. https://doi.org/10.1007/978-3-642-25878-7_36

}

TY - GEN

T1 - Monotone drawings of graphs with fixed embedding

AU - Angelini, Patrizio

AU - Didimo, Walter

AU - Kobourov, Stephen G

AU - McHedlidze, Tamara

AU - Roselli, Vincenzo

AU - Symvonis, Antonios

AU - Wismath, Stephen

PY - 2012

Y1 - 2012

N2 - 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, and we show that biconnected embedded planar graphs and outerplane graphs always admit such drawings, which can be computed in linear time.

AB - 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, and we show that biconnected embedded planar graphs and outerplane graphs always admit such drawings, which can be computed in linear time.

UR - http://www.scopus.com/inward/record.url?scp=84455209585&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84455209585&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-25878-7_36

DO - 10.1007/978-3-642-25878-7_36

M3 - Conference contribution

AN - SCOPUS:84455209585

SN - 9783642258770

VL - 7034 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 379

EP - 390

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -