Abstract
We investigate the complexity of some problems related to the Simultaneous Embedding with Fixed Edges (SEFE) problem which, given k planar graphs G 1,…,G k on the same set of vertices, asks whether they can be simultaneously embedded so that the embedding of each graph be planar and common edges be drawn the same. While the computational complexity of SEFE with k = 2 is a central open question in Graph Drawing, the problem is \(\mathcal{NP}\)-complete for k ≥ 3 [Gassner et al., WG ’06], even if the intersection graph is the same for each pair of graphs (sunflower intersection) [Schaefer, JGAA (2013)].
We improve on these results by proving that SEFE with k ≥ 3 and sunflower intersection is \(\mathcal{NP}\)-complete even when (i) the intersection graph is connected and (ii) two of the three input graphs are biconnected. This result implies that the Partitioned T-Coherent k-Page Book-Embedding is \(\mathcal{NP}\)-complete with k ≥ 3, which was only known for k unbounded [Hoske, Bachelor Thesis (2012)]. Further, we prove that the problem of maximizing the number of edges that are drawn the same in a SEFE of two graphs is \(\mathcal{NP}\)-complete (optimization of SEFE, Open Problem 9, Chapter 11 of the Handbook of Graph Drawing and Visualization).
Part of the research was conducted in the framework of ESF project 10-EuroGIGA-OP-003 GraDR ”Graph Drawings and Representations” and of ’EU FP7 STREP Project ”Leone: From Global Measurements to Local Management”, grant no. 317647’.
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
Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F., Roselli, V.: Beyond clustered planarity. CoRR abs/1207.3934 (2012)
Angelini, P., Da Lozzo, G., Neuwirth, D.: On the complexity of some problems related to SEFE. CoRR abs/1311.3607 (2013)
Angelini, P., Di Battista, G., Frati, F.: Simultaneous embedding of embedded planar graphs. Int. J. on Comp. Geom. and Appl. (2013), Special Issue from Asano, T., Nakano, S.-I., Okamoto, Y., Watanabe, O. (eds.): ISAAC 2011. LNCS, vol. 7074. Springer, Heidelberg (2011)
Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. In: SODA 2010, pp. 202–221 (2010)
Angelini, P., Di Battista, G., Frati, F., Patrignani, M., Rutter, I.: Testing the simultaneous embeddability of two graphs whose intersection is a biconnected or a connected graph. J. of Discrete Algorithms 14, 150–172 (2012)
Blasiüs, T., Kobourov, S.G., Rutter, I.: Simultaneous embedding of planar graphs. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization. CRC Press (2013)
Bläsius, T., Rutter, I.: Disconnectivity and relative positions in simultaneous embeddings. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 31–42. Springer, Heidelberg (2013)
Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. In: SODA 2013, pp. 1030–1043 (2013)
Di Battista, G., Frati, F.: Efficient c-planarity testing for embedded flat clustered graphs with small faces. J. of Graph Alg. and App. 13(3), 349–378 (2009), Special Issue from Hong, S.-H., Nishizeki, T., Quan, W. (eds.): GD 2007. LNCS, vol. 4875. Springer, Heidelberg (2008)
Eades, P., Feng, Q.W., Lin, X., Nagamochi, H.: Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Algorithmica 44(1), 1–32 (2006)
Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. J. of Graph Alg. and Appl. 9(3), 347–364 (2005)
Erten, C., Kobourov, S.G., Le, V., Navabi, A.: Simultaneous graph drawing: Layout algorithms and visualization schemes. J. of Graph Alg. and Appl. 9(1), 165–182 (2005)
Garey, M.R., Johnson, D.S.: The Rectilinear Steiner Tree Problem is NP-Complete. SIAM J. Appl. Math. 32, 826–834 (1977)
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)
Haeupler, B., Jampani, K.R., Lubiw, A.: Testing simultaneous planarity when the common graph is 2-connected. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part II. LNCS, vol. 6507, pp. 410–421. Springer, Heidelberg (2010)
Hoske, D.: Book embedding with fixed page assignments. Bachelor thesis, Karlsruhe Institute of Technology, Karlsruhe, Germany (2012)
Jünger, M., Schulz, M.: Intersection graphs in simultaneous embedding with fixed edges. J. of Graph Alg. and Appl. 13(2), 205–218 (2009)
Moore, C., Mertens, S.: The Nature of Computation. Oxford University Press, USA (2011)
Opatrny, J.: Total ordering problem. SIAM J. Comput. 8(1), 111–114 (1979)
Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17(4), 717–728 (2001)
Schaefer, M.: Toward a theory of planarity: Hanani-tutte and planarity variants. J. of Graph Alg. and Appl. 17(4), 367–440 (2013)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Angelini, P., Da Lozzo, G., Neuwirth, D. (2014). On Some \(\mathcal{NP}\)-complete SEFE Problems. In: Pal, S.P., Sadakane, K. (eds) Algorithms and Computation. WALCOM 2014. Lecture Notes in Computer Science, vol 8344. Springer, Cham. https://doi.org/10.1007/978-3-319-04657-0_20
Download citation
DOI: https://doi.org/10.1007/978-3-319-04657-0_20
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-04656-3
Online ISBN: 978-3-319-04657-0
eBook Packages: Computer ScienceComputer Science (R0)