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.
Research partially supported by the MIUR project AlgoDEEP prot. 2008TFBWL4, by the ESF project 10-EuroGIGA-OP-003 GraDR “Graph Drawings and Representations”, by NSERC, and by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning’” of the National Strategic Reference Framework (NSRF) - Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund. Work on these results began at the 6th Bertinoro Workshop on Graph drawing. Discussion with other participants is gratefully acknowledged.
Chapter PDF
Similar content being viewed by others
References
Angelini, P., Colasante, E., Di Battista, G., Frati, F., Patrignani, M.: Monotone Drawings of Graphs. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 13–24. Springer, Heidelberg (2011)
Angelini, P., Frati, F., Grilli, L.: An algorithm to construct greedy drawings of triangulations. J. Graph Algorithms Appl. 14(1), 19–51 (2010)
Arkin, E.M., Connelly, R., Mitchell, J.S.B.: On monotone paths among obstacles with applications to planning assemblies. In: Symposium on Computational Geometry, pp. 334–343 (1989)
Brocot, A.: Calcul des rouages par approximation, nouvelle methode. Revue Chronometrique 6, 186–194 (1860)
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999)
Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. Comput. 25, 956–997 (1996)
Garg, A., Tamassia, R.: Upward planarity testing. Order 12, 109–133 (1995)
Huang, W., Eades, P., Hong, S.-H.: A graph reading behavior: Geodesic-path tendency. In: PacificVis, pp. 137–144 (2009)
Leighton, T., Moitra, A.: Some results on greedy embeddings in metric spaces. Discrete & Computational Geometry 44(3), 686–705 (2010)
Papadimitriou, C.H., Ratajczak, D.: On a conjecture related to geometric routing. Theor. Comput. Sci. 344(1), 3–14 (2005)
Stern, M.A.: Ueber eine zahlentheoretische funktion. Journal fur die reine und angewandte Mathematik 55, 193–220 (1858)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Angelini, P. et al. (2012). Monotone Drawings of Graphs with Fixed Embedding. In: van Kreveld, M., Speckmann, B. (eds) Graph Drawing. GD 2011. Lecture Notes in Computer Science, vol 7034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25878-7_36
Download citation
DOI: https://doi.org/10.1007/978-3-642-25878-7_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25877-0
Online ISBN: 978-3-642-25878-7
eBook Packages: Computer ScienceComputer Science (R0)