Abstract
In this paper, we study the geometric RAC simultaneous drawing problem: Given two planar graphs that share a common vertex set but have disjoint edge sets, a geometric RAC simultaneous drawing is a straight-line drawing in which (i) each graph is drawn planar, (ii) there are no edge overlaps, and, (iii) crossings between edges of the two graphs occur at right-angles. We first prove that two planar graphs admitting a geometric simultaneous drawing may not admit a geometric RAC simultaneous drawing. We further show that a cycle and a matching always admit a geometric RAC simultaneous drawing, which can be constructed in linear time.
We also study a closely related problem according to which we are given a planar embedded graph G and the main goal is to determine a geometric drawing of G and its dual G * (without the face-vertex corresponding to the external face) such that: (i) G and G * are drawn planar, (ii) each vertex of the dual is drawn inside its corresponding face of G and, (iii) the primal-dual edge crossings form right-angles. We prove that it is always possible to construct such a drawing if the input graph is an outerplanar embedded graph.
The work of E.N. Argyriou has been co-financed 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. 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.
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References
Angelini, P., Cittadini, L., Di Battista, G., Didimo, W., Frati, F., Kaufmann, M., Symvonis, A.: On the perspectives opened by right angle crossing drawings. JGAA 15(1), 53–78 (2011)
Angelini, P., Geyer, M., Kaufmann, M., Neuwirth, D.: On a Tree and a Path with No Geometric Simultaneous Embedding. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 38–49. Springer, Heidelberg (2011)
Argyriou, E.N., Bekos, M.A., Symvonis, A.: The Straight-Line RAC Drawing Problem Is NP-Hard. In: Černá, I., Gyimóthy, T., Hromkovič, J., Jefferey, K., Králović, R., Vukolić, M., Wolf, S. (eds.) SOFSEM 2011. LNCS, vol. 6543, pp. 74–85. Springer, Heidelberg (2011)
Arikushi, K., Fulek, R., Keszegh, B., Morić, F., Tóth, C.D.: Graphs that Admit Right Angle Crossing Drawings. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 135–146. Springer, Heidelberg (2010)
Brass, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Computational Geometry: Theory and Applications 36(2), 117–130 (2007)
Brightwell, G., Scheinerman, E.R.: Representations of planar graphs. SIAM Journal Discrete Mathematics 6(2), 214–229 (1993)
Cabello, S., van Kreveld, M.J., Liotta, G., Meijer, H., Speckmann, B., Verbeek, K.: Geometric simultaneous embeddings of a graph and a matching. JGAA 15(1), 79–96 (2011)
Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H.: Area, Curve Complexity, and Crossing Resolution of Non-planar Graph Drawings. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 15–20. Springer, Heidelberg (2010)
Didimo, W., Eades, P., Liotta, G.: A characterization of complete bipartite graphs. Information Processing Letters 110(16), 687–691 (2010)
Didimo, W., Eades, P., Liotta, G.: Drawing graphs with right angle crossings. Theor. Comp. Sci. 412(39), 5156–5166 (2011)
Eades, P., Liotta, G.: Right angle crossing graphs and 1-planarity. In: 27th European Workshop on Computational Geometry (2011)
Erten, C., Kobourov, S.G.: Simultaneous embedding of a planar graph and its dual on the grid. Theory Computing 38(3), 313–327 (2005)
Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. JGAA 9(3), 347–364 (2005)
Estrella-Balderrama, A., Gassner, E., Jünger, M., Percan, M., Schaefer, M., Schulz, M.: Simultaneous Geometric Graph Embeddings. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 280–290. Springer, Heidelberg (2008)
Frati, F., Kaufmann, M., Kobourov, S.G.: Constrained simultaneous and near-simultaneous embeddings. JGAA 13(3), 447–465 (2009)
Geyer, M., Kaufmann, M., Vrto, I.: Two trees which are self-intersecting when drawn simultaneously. Discrete Mathematics 309(7), 1909–1916 (2009)
Huang, W., Hong, S.H., Eades, P.: Effects of crossing angles. In: IEEE Pacific Visualization Symp., pp. 41–46. IEEE (2008)
van Kreveld, M.: The Quality Ratio of RAC Drawings and Planar Drawings of Planar Graphs. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 371–376. Springer, Heidelberg (2011)
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Argyriou, E., Bekos, M., Kaufmann, M., Symvonis, A. (2012). Geometric RAC Simultaneous Drawings of Graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_25
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DOI: https://doi.org/10.1007/978-3-642-32241-9_25
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