Algorithmica

, Volume 71, Issue 2, pp 233–257

# Monotone Drawings of Graphs with Fixed Embedding

• Patrizio Angelini
• Walter Didimo
• Stephen Kobourov
• Tamara Mchedlidze
• Vincenzo Roselli
• Antonios Symvonis
• Stephen Wismath
Article

## 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.

## Keywords

Monotone drawings Fixed embedding Planar graph drawing Polynomial area Curve complexity

## Notes

### Acknowledgements

We thank the anonymous reviewers for their valuable comments.

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© Springer Science+Business Media New York 2013

## Authors and Affiliations

• Patrizio Angelini
• 1
• Walter Didimo
• 2
• Stephen Kobourov
• 3
• Tamara Mchedlidze
• 4
• Vincenzo Roselli
• 1
• Antonios Symvonis
• 5
• Stephen Wismath
• 6
1. 1.Università Roma TreRomaItaly
2. 2.Università degli Studi di PerugiaPerugiaItaly
3. 3.University of ArizonaTucsonUSA
4. 4.Faculty of InformaticsKarlsruhe Institute of Technology (KIT)KarlsruheGermany
5. 5.National Technical University of AthensAthensGreece
6. 6.University of LethbridgeLethbridgeCanada