Drawing Partially Embedded and Simultaneously Planar Graphs

Chan, Timothy M. and Frati, Fabrizio and Gutwenger, Carsten and Lubiw, Anna and Mutzel, Petra and Schaefer, Marcus (2014) Drawing Partially Embedded and Simultaneously Planar Graphs. In: Graph Drawing 22nd International Symposium, GD 2014, September 24-26, 2014, Würzburg, Germany , pp. 25-39 (Official URL: http://dx.doi.org/10.1007/978-3-662-45803-7_3).

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Abstract

We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph (PEG) problem—to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph—and the simultaneous planarity (SEFE) problem—to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required planar drawings exist, then there are planar drawings with a linear number of bends per edge and, in the case of simultaneous planarity, a constant number of crossings between every pair of edges. Our proofs provide efficient algorithms if the combinatorial embedding information about the drawing is given. Our result on partially embedded graph drawing generalizes a classic result of Pach and Wenger showing that any planar graph can be drawn with fixed locations for its vertices and with a linear number of bends per edge.

Item Type:Conference Paper
Additional Information:10.1007/978-3-662-45803-7_3
Classifications:G Algorithms and Complexity > G.210 Bends
G Algorithms and Complexity > G.420 Crossings
G Algorithms and Complexity > G.490 Embeddings
P Styles > P.540 Planar
ID Code:1419

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References

Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. In: Proc. Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 202–221. SIAM (2010)

Badent, M., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. Theor. Comput. Sci. 408(2-3), 129–142 (2008)

Bern, M., Gilbert, J.R.: Drawing the planar dual. Inform. Process. Lett. 43(1), 7–13 (1992)

Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embeddings of planar graphs. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization. Discrete Mathematics and Its Applications, ch. 11, pp. 349–382. Chapman and Hall/CRC (2013)

Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM Journal on Computing 42(5), 1803–1829 (2013)

Chan, T.M., Hoffmann, H.-F., Kiazyk, S., Lubiw, A.: Minimum length embedding of planar graphs at fixed vertex locations. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 376–387. Springer, Heidelberg (2013)

Chimani, M., Jünger, M., Schulz, M.: Crossing minimization meets simultaneous drawing. In: PacificVis, pp. 33–40. IEEE (2008)

Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. J. Graph Algorithms and Appl. 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)

Fowler, J.J., Jünger, M., Kobourov, S.G., Schulz, M.: Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges. Comput. Geom. 44(8), 385–398 (2011)

Gassner, E., Jünger, M., Percan, M., Schaefer, M., Schulz, M.: Simultaneous graph embeddings with fixed edges. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 325–335. Springer, Heidelberg (2006)

Grilli, L., Hong, S.-H., Kratochvíl, J., Rutter, I.: Drawing simultaneously embedded graphs with few bends. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 40–51. Springer, Heidelberg (2014)

Haeupler, B., Jampani, K.R., Lubiw, A.: Testing simultaneous planarity when the common graph is 2-connected. J. Graph Algorithms and Appl. 17(3), 147–171 (2013)

Hliněný, P.: Crossing number is hard for cubic graphs. J. Combin. Theory Ser. B 96(4), 455–471 (2006)

Jelínek, V., Kratochvíl, J., Rutter, I.: A Kuratowski-type theorem for planarity of partially embedded graphs. Comput. Geom. 46(4), 466–492 (2013)

Jünger, M., Schulz, M.: Intersection graphs in simultaneous embedding with fixed edges. J. Graph Algorithms Appl. 13(2), 205–218 (2009)

Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms and Appl. 6(1), 115–129 (2002)

Kratochvíl, J., Matoušek, J.: String graphs requiring exponential representations. J. Comb. Theory, Ser. B 53(1), 1–4 (1991)

O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press, NY (1987)

Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs Combin. 17(4), 717–728 (2001)

Patrignani, M.: On extending a partial straight-line drawing. Internat. J. Found. Comput. Sci. 17(5), 1061–1069 (2006)

Schaefer, M.: The graph crossing number and its variants: A survey. The Electronic Journal of Combinatorics 20, 1–90 (2013), Dynamic Survey, #DS21.

Schaefer, M.: Toward a theory of planarity: Hanani-Tutte and planarity variants. J. of Graph Algorthims and Appl. 17(4), 367–440 (2013)