Abstract
A 2-page drawing of a graph is such that the vertices are drawn as points along a line and each edge is a circular arc in one of the two half-planes defined by this line. If all edges are in the same half-plane, the drawing is called a 1-page drawing. We want to compute 1-page and 2-page drawings of planar graphs such that the number of crossings per edge does not depend on the number of the vertices. We show that for any constant k, there exist planar graphs that require more than k crossings per edge in either a 1-page or a 2-page drawing. We then prove that if the vertex degree is bounded by \(\varDelta \), every planar 3-tree has a 2-page drawing with a number of crossings per edge that only depends on \(\varDelta \). Finally, we show a similar result for 1-page drawings of partial 2-trees.
Research supported in part by the MIUR project AMANDA “Algorithmics for MAssive and Networked DAta”, prot. 2012C4E3KT_001.
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Notes
- 1.
A multigraph is a graph that can have multiple edges between the same pair of vertices.
References
Alam, M.J., Brandenburg, F.J., Kobourov, S.G.: Straight-line grid drawings of 3-connected 1-planar graphs. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 83–94. Springer, Heidelberg (2013)
Bannister, M.J., Eppstein, D.: Crossing minimization for 1-page and 2-page drawings of graphs with bounded treewidth. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 210–221. Springer, Heidelberg (2014)
Bannister, M.J., Eppstein, D., Simons, J.A.: Fixed parameter tractability of crossing minimization of almost-trees. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 340–351. Springer, Heidelberg (2013)
Bernhart, F., Kainen, P.C.: The book thickness of a graph. J. Comb. Theor. Ser. B 27(3), 320–331 (1979)
Bodlaender, H.L.: A partial \(k\)-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209(12), 1–45 (1998)
Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Embedding graphs in books: a layout problem with applications to VLSI design. SIAM J. Discrete Math. 8(1), 33–58 (1987)
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Upper Saddle River (1999)
Di Giacomo, E., Didimo, W., Liotta, G., Montecchiani, F.: Area requirement of graph drawings with few crossings per edge. Comput. Geom. 46(8), 909–916 (2013)
Di Giacomo, E., Liotta, G., Montecchiani, F.: Drawing outer 1-planar graphs with few slopes. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 174–185. Springer, Heidelberg (2014)
Masuda, S., Nakajima, K., Kashiwabara, T., Fujisawa, T.: Crossing minimization in linear embeddings of graphs. IEEE Trans. Comput. 39(1), 124–127 (1990)
Nicholson, T.: Permutation procedure for minimising the number of crossings in a network. Proc. Inst. Electr. Eng. 115(1), 21–26 (1968)
Wattenberg, M.: Arc diagrams: visualizing structure in strings. In: InfoVis 2002, pp. 110–116. IEEE Computer Society (2002)
Yannakakis, M.: Embedding planar graphs in four pages. J. Comput. Syst. Sci. 38(1), 36–67 (1989)
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Binucci, C., Di Giacomo, E., Hossain, M.I., Liotta, G. (2016). 1-Page and 2-Page Drawings with Bounded Number of Crossings per Edge. In: Lipták, Z., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2015. Lecture Notes in Computer Science(), vol 9538. Springer, Cham. https://doi.org/10.1007/978-3-319-29516-9_4
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DOI: https://doi.org/10.1007/978-3-319-29516-9_4
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