Abstract
In smooth orthogonal layouts of planar graphs, every edge is an alternating sequence of axis-aligned segments and circular arcs with common axis-aligned tangents. In this paper, we study the problem of finding smooth orthogonal layouts of low edge complexity, that is, with few segments per edge. We say that a graph has smooth complexity k—for short, an SC k -layout—if it admits a smooth orthogonal drawing of edge complexity at most k.
Our main result is that every 4-planar graph has an SC2-layout. While our drawings may have super-polynomial area, we show that for 3-planar graphs, cubic area suffices. We also show that any biconnected 4-outerplane graph has an SC1-layout. On the negative side, we demonstrate an infinite family of biconnected 4-planar graphs that require exponential area for an SC1-layout. Finally, we present an infinite family of biconnected 4-planar graphs that do not admit an SC1-layout.
Research of M.J. Alam and S.G. Kobourov is supported in part by NSF grants CCF-1115971 and DEB 1053573. The work of M.A. Bekos is implemented within the framework of the Action “Supporting Postdoctoral Researchers” of the Operational Program “Education and Lifelong Learning” (Action’s Beneficiary: General Secretariat for Research and Technology), and is co-financed by the European Social Fund (ESF) and the Greek State. M. Kaufmann as well as Ph. Kindermann and A. Wolff acknowledge support by the ESF EuroGIGA project GraDR (DFG grants Ka 812/16-1 and Wo 758/5-1, respectively).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alam, M.J., Bekos, M.A., Kaufmann, M., Kindermann, P., Kobourov, S.G., Wolff, A.: Smooth orthogonal drawings of planar graphs. Arxiv report arxiv.org/abs/1312.3538 (2013)
Bekos, M.A., Kaufmann, M., Kobourov, S.G., Symvonis, A.: Smooth orthogonal layouts. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 150–161. Springer, Heidelberg (2013)
Biedl, T., Kant, G.: A better heuristic for orthogonal graph drawings. Comput. Geom. Theory Appl. 9(3), 159–180 (1998)
Duncan, C.A., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Nöllenburg, M.: Lombardi drawings of graphs. J. Graph Algorithms Appl. 16(1), 85–108 (2012), http://dx.doi.org/10.7155/jgaa.00251
Duncan, C.A., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Nöllenburg, M.: Drawing trees with perfect angular resolution and polynomial area. Discrete Comput. Geom. 49(2), 157–182 (2013)
Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2001)
Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996)
Leiserson, C.E.: Area-efficient graph layouts (for VLSI). In: Proc. 21st Annu. IEEE Symp. Foundat. Comput. Sci. (FOCS 1980), pp. 270–281 (1980)
Liu, Y., Morgana, A., Simeone, B.: A linear algorithm for 2-bend embeddings of planar graphs in the two-dimensional grid. Discrete Appl. Math. 81(1-3), 69–91 (1998)
Seemann, J.: Extending the Sugiyama algorithm for drawing UML class diagrams: Towards automatic layout of object-oriented software diagrams. In: Di Battista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 415–424. Springer, Heidelberg (1997)
Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)
Waldherr, H.: Network Gmunden–Vöcklabruck–Salzkammergut of the Publ. Transp. Assoc. OÖVG, Austria (November 2012), http://www.ooevv.at/uploads/media/OOE2_Salzkammergut_V17_END.pdf (accessed September 10, 2013)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Alam, M.J., Bekos, M.A., Kaufmann, M., Kindermann, P., Kobourov, S.G., Wolff, A. (2014). Smooth Orthogonal Drawings of Planar Graphs. In: Pardo, A., Viola, A. (eds) LATIN 2014: Theoretical Informatics. LATIN 2014. Lecture Notes in Computer Science, vol 8392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54423-1_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-54423-1_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54422-4
Online ISBN: 978-3-642-54423-1
eBook Packages: Computer ScienceComputer Science (R0)