Abstract
In this paper, we study small planar drawings of planar graphs. For arbitrary planar graphs, Θ(n 2) is the established upper and lower bound on the worst-case area. It is a long-standing open problem for what graphs smaller area can be achieved, with results known only for trees and outer-planar graphs. We show here that series-parallel can be drawn in O(n 3/2) area, but 2-outer-planar graphs and planar graphs of proper pathwidth 3 require Ω(n 2) area.
Research supported by NSERC. Part of the work was done while the author was on sabbatical leave at University of Passau.
Chapter PDF
Similar content being viewed by others
References
Bertolazzi, P., Cohen, R.F., Di Battista, G., Tamassia, R., Tollis, I.G.: How to draw a series-parallel digraph. Intl. J. Comput. Geom. Appl. 4, 385–402 (1994)
Biedl, T.: New lower bounds for orthogonal graph drawings. Journal of Graph Algorithms and Applications 2(7), 1–31 (1998)
Biedl, T.: Drawing outer-planar graphs in O(nlogn) area. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 54–65. Springer, Heidelberg (2002)
Bodlaender, H.: Treewidth: algorithmic techniques and results. In: Privara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 19–36. Springer, Heidelberg (1997)
Brandenburg, F., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Liotta, G., Mutzel, P.: Selected open problems in graph drawing. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 515–539. Springer, Heidelberg (2004)
Cohen, R., Di Battista, G., Tamassia, R., Tollis, I.: Dynamic graph drawings: Trees, series-parallel digraphs, and planar st-digraphs. SIAM J. Comput. 24(5), 970–1001 (1995)
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.: Graph Drawing: Algorithms for Geometric Representations of Graphs. Prentice-Hall, Englewood Cliffs (1998)
Di Battista, G., Frati, F.: Small area drawings of outerplanar graphs. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 89–100. Springer, Heidelberg (2006)
Dujmovic, V., Fellows, M., Kitching, M., Liotta, G., McCartin, C., Nishimura, N., Ragde, P., Rosamond, F., Whitesides, S., Wood, D.: On the parameterized complexity of layered graph drawing. Algorithmica 52, 267–292 (2008)
Dujmovic, V., Morin, P., Wood, D.K.: Layout of graphs with bounded tree-width. SIAM J. on Computing 34(3), 553–579 (2005)
Felsner, S., Liotta, G., Wismath, S.: Straight-line drawings on restricted integer grids in two and three dimensions. Journal of Graph Algorithms and Applications 7(4), 335–362 (2003)
Frati, F.: Straight-line drawings of outerplanar graphs in o(dn log n) area. In: Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG 2007), pp. 225–228 (2007)
Frati, F.: A lower bound on the area requirements of series-parallel graphs. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 159–170. Springer, Heidelberg (2008)
de Fraysseix, H., Pach, J., Pollack, R.: Small sets supporting fary embeddings of planar graphs. In: Twentieth Annual ACM Symposium on Theory of Computing, pp. 426–433 (1988)
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)
Hong, S.H., Eades, P., Quigley, A., Lee, S.H.: Drawing algorithms for series-parallel digraphs in two and three dimensions. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 198–209. Springer, Heidelberg (1999)
Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16, 4–32 (1996)
Leiserson, C.: Area-efficient graph layouts (for VLSI). In: 21st IEEE Symposium on Foundations of Computer Science, pp. 270–281 (1980)
Miura, K., Nishizeki, T., Nakano, S.: Grid drawings of 4-connected plane graphs. Discrete Computational Geometry 26, 73–87 (2001)
Schnyder, W.: Embedding planar graphs on the grid. In: 1st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 138–148 (1990)
Tamassia, R., Tollis, I.: A unified approach to visibility representations of planar graphs. Discrete Computational Geometry 1, 321–341 (1986)
Tayu, S., Nomura, K., Ueno, S.: On the two-dimensional orthogonal drawing of series-parallel graphs. Discr. Appl. Mathematics 157(8), 1885–1895 (2009)
Wismath, S.: Characterizing bar line-of-sight graphs. In: 1st ACM Symposium on Computational Geometry, Baltimore, Maryland, USA, pp. 147–152 (1985)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Biedl, T. (2010). Small Drawings of Series-Parallel Graphs and Other Subclasses of Planar Graphs. In: Eppstein, D., Gansner, E.R. (eds) Graph Drawing. GD 2009. Lecture Notes in Computer Science, vol 5849. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11805-0_27
Download citation
DOI: https://doi.org/10.1007/978-3-642-11805-0_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11804-3
Online ISBN: 978-3-642-11805-0
eBook Packages: Computer ScienceComputer Science (R0)