Abstract
We present a linear-time algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G ∖ C is contained entirely inside or outside C? For the latter problem, we present an algorithm that is based on SPQR-trees and has linear running time. We also show how we can employ SPQR-trees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQR-tree approach might eventually lead to a polynomial-time algorithm for deciding the general SEFE problem for two planar graphs.
Chapter PDF
References
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)
Di Battista, G., Didimo, W., Patrignani, M., Pizzonia, M.: Drawing database schemas. Software: Practice and Experience 32(11), 1065–1098 (2002)
Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM Journal on Computing 25(5), 956–997 (1996)
Di Giacomo, E., Liotta, G.: A note on simultaneous embedding of planar graphs. In: EWCG 2005, pp. 207–210 (2005)
Dornheim, C.: Planar graphs with topological constraints. Journal on Graph Algorithms and Applications 6(1), 27–66 (2002)
Eiglsperger, M., Fößmeier, U., Kaufmann, M.: Orthogonal graph drawing with constraints. In: Proc. SODA 2000, pp. 3–11 (2000)
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)
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. In: WG 2008 (to appear)
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)
Gutwenger, C., Klein, K., Mutzel, P.: Planarity testing and optimal edge insertion with embedding constraints. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 126–137. Springer, Heidelberg (2007)
Hershberger, J., Suri, S.: An optimal algorithm for Euclidean shortest paths in the plane. SIAM Journal on Computing 28(6), 2215–2256 (1999)
Veblen, O.: Theory on plane curves in non-metrical analysis situs. Transactions of the American Mathematical Society 6, 83–98 (1905)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fowler, J.J., Gutwenger, C., Jünger, M., Mutzel, P., Schulz, M. (2009). An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges. In: Tollis, I.G., Patrignani, M. (eds) Graph Drawing. GD 2008. Lecture Notes in Computer Science, vol 5417. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00219-9_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-00219-9_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00218-2
Online ISBN: 978-3-642-00219-9
eBook Packages: Computer ScienceComputer Science (R0)