Abstract
A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open.
We give a necessary and sufficient condition for whether a SEFE exists for pairs of graphs whose union is homeomorphic to K 5 or K 3,3. This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide efficient algorithms to compute a SEFE. Finally, we provide a linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has a SEFE.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Brandes, U., Erten, C., Fowler, J., Frati, F., Geyer, M., Gutwenger, C., Hong, S., Kaufmann, M., Kobourov, S., Liotta, G., Mutzel, P., Symvonis, A.: Colored simultaneous geometric embeddings. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 254–263. Springer, Heidelberg (2007)
Brass, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous graph embedding. Computational Geometry 36(2), 117–130 (2007)
Chiba, N., Nishizeki, T., Abe, S., Ozawa, T.: A linear algorithm for embedding planar graphs using PQ-trees. J. Comput. Syst. Sci. 30(1), 54–76 (1985)
Chimani, M., Jünger, M., Schulz, M.: Crossing minimization meets simultaneous drawing. In: IEEE Pacific Visualization Symposium 2008, pp. 33–40 (2008)
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)
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. Technical Report TR08-01, University of Arizona (2008), ftp://ftp.cs.arizona.edu/reports/2008/TR08-01.pdf
Frati, F.: Embedding graphs simultaneously with fixed edges. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 108–113. Springer, Heidelberg (2007)
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)
Geyer, M., Kaufmann, M., Vrto, 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)
Hershberger, J., Suri, S.: An optimal algorithm for Euclidean shortest paths in the plane. SIAM Journal on Computing 28(6), 2215–2256 (1999)
Kuratowski, C.: Sur les problèmes des courbes gauches en Topologie. Fund. Math. 15, 271–283 (1930)
Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 263–274. Springer, Heidelberg (1998)
Wagner, K.: Über eine Eigenschaft der ebenen Komplexe. Math. Ann. 114(1), 570–590 (1937)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fowler, J.J., Jünger, M., Kobourov, S., Schulz, M. (2008). Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2008. Lecture Notes in Computer Science, vol 5344. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92248-3_14
Download citation
DOI: https://doi.org/10.1007/978-3-540-92248-3_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92247-6
Online ISBN: 978-3-540-92248-3
eBook Packages: Computer ScienceComputer Science (R0)