Abstract
We study the problem of simultaneously embedding several graphs on the same vertex set in such a way that edges common to two or more graphs are represented by the same curve. This problem is known as simultaneously embedding graphs with fixed edges. We show that this problem is closely related to the weak realizability problem: Can a graph be drawn such that all edge crossings occur in a given set of edge pairs? By exploiting this relationship we can explain why the simultaneous embedding problem is challenging, both from a computational and a combinatorial point of view.
More precisely, we prove that simultaneously embedding graphs with fixed edges is NP-complete even for three planar graphs. For two planar graphs the complexity status is still open.
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
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. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 243–255. Springer, Heidelberg (2003)
Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 195–205. Springer, Heidelberg (2005)
Erten, C., Kobourov, S.G.: Simultaneous embedding of a planar graph and its dual on the grid. Theory of Computing Systems 38, 313–327 (2005)
Erten, C., Kobourov, S.G., Le, V., Navabi, A.: Simultaneous graph drawing: Layout algorithms and visualization schemes. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 437–449. Springer, Heidelberg (2004)
Geyer, M., Kaufmann, M., Vrt’o, I.: Two trees which are self–intersecting when drawn simultaneously. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 201–210. Springer, Heidelberg (2006)
Di Giacomo, E., Liotta, G.: A note on simultaneous embedding of planar graphs. In: 21st European Workshop on Comp. Geometry, pp. 207–210 (2005)
Kobourov, S.G., Pitta, C.: An interactive multi-user system for simultaneous graph drawing. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 492–501. Springer, Heidelberg (2005)
Kratochvíl, J.: String graphs II Recognizing string graphs is NP-hard. Journal of Combinatorial Theory B 52, 67–78 (1991)
Kratochvíl, J.: Crossing number of abstract topological graphs. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 238–245. Springer, Heidelberg (1999)
Kratochvíl, J., Lubiw, A., Nešetřil, J.: Noncrossing subgraphs in topological layouts. SIAM Journal on Discrete Mathematics 4(2), 223–244 (1991)
Schaefer, M., Sedgwick, E., Štefankovič, D.: Recognizing string graphs in NP. Journal Comput. Syst. Sci. 67(2), 365–380 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gassner, E., Jünger, M., Percan, M., Schaefer, M., Schulz, M. (2006). Simultaneous Graph Embeddings with Fixed Edges. In: Fomin, F.V. (eds) Graph-Theoretic Concepts in Computer Science. WG 2006. Lecture Notes in Computer Science, vol 4271. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11917496_29
Download citation
DOI: https://doi.org/10.1007/11917496_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-48381-6
Online ISBN: 978-3-540-48382-3
eBook Packages: Computer ScienceComputer Science (R0)