ISAAC 2010: Algorithms and Computation pp 410-421

# Testing Simultaneous Planarity When the Common Graph Is 2-Connected

• Bernhard Haeupler
• Krishnam Raju Jampani
• Anna Lubiw
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6507)

## Abstract

Two planar graphs G 1 and G 2 sharing some vertices and edges are simultaneously planar if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a linear-time algorithm to test simultaneous planarity when the two graphs share a 2-connected subgraph. Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them.

## Keywords

Simultaneous Embedding Planar Graph PQ Tree Graph Drawing

## References

1. 1.
Angelini, P., Di Battista, G., Frati, F., Patrignani, M., Rutter, I.: Testing the simultaneous embeddability of two graphs whose intersection is a biconnected graph or a tree. In: IWOCA. LNCS (2010)Google Scholar
2. 2.
Angelini, P., Geyer, M., Kaufmann, M., Neuwirth, D.: On a tree and a path with no geometric simultaneous embedding. In: CoRR, abs/1001.0555 (2010)Google Scholar
3. 3.
Booth, K.: PQ Tree Algorithms. PhD thesis, University of California, Berkeley (1975)Google Scholar
4. 4.
Booth, K., Lueker, G.: Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. Journal of Computer and System Sciences 13, 335–379 (1976)
5. 5.
Boyer, J., Myrvold, W.: On the cutting edge: Simplified O(n) planarity by edge addition. Journal of Graph Algorithms and Applications 8(3), 241–273 (2004)
6. 6.
Brass, P., Cenek, E., Duncan, C., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.: On simultaneous planar graph embeddings. Computational Geometry: Theory and Applications 36(2), 117–130 (2007)
7. 7.
Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., Schrijver, A.: Combinatorial Optimization. Wiley Interscience, Hoboken (1997)
8. 8.
DiGiacomo, G., Liotta, G.: Simultaneous embedding of outerplanar graphs, paths, and cycles. International Journal of Computational Geometry and Applications 17(2), 139–160 (2007)
9. 9.
Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. Journal of Graph Algorithms and Applications 9(3), 347–364 (2005)
10. 10.
Estrella-Balderrama, A., Gassner, E., Junger, 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)
11. 11.
Even, S., Tarjan, R.: Computing an st-Numbering. Theor. Comput. Sci. 2(3), 339–344 (1976)
12. 12.
Fowler, J., Gutwenger, C., Junger, M., Mutzel, P., Schulz, M.: An SPQR-tree approach to decide special cases of simultaneous embedding with fixed edges. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 1–12. Springer, Heidelberg (2009)
13. 13.
Fowler, J., Jünger, M., Kobourov, S.G., Schulz, M.: Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 146–158. Springer, Heidelberg (2008)
14. 14.
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)
15. 15.
Gassner, E., Junger, 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)
16. 16.
Haeupler, B., Jampani, K.R., Lubiw, A.: Testing simultaneous planarity when the common graph is 2-connected (2010)Google Scholar
17. 17.
Haeupler, B., Tarjan, R.E.: Planarity algorithms via PQ-trees (extended abstract). Electronic Notes in Discrete Mathematics 31, 143–149 (2008)
18. 18.
Jampani, K.R., Lubiw, A.: The simultaneous representation problem for chordal, comparability and permutation graphs. In: WADS. LNCS, vol. 5664, pp. 387–398. Springer, Heidelberg (2009)Google Scholar
19. 19.
Jampani, K.R., Lubiw, A.: Simultaneous interval graphs (2010) (submitted)Google Scholar
20. 20.
Jünger, M., Leipert, S.: Level planar embedding in linear time. J. Graph Algorithms Appl. 6(1), 67–113 (2002)
21. 21.
Jünger, M., Schulz, M.: Intersection graphs in simultaneous embedding with fixed edges. Journal of Graph Algorithms and Applications 13(2), 205–218 (2009)
22. 22.
Lempel, A., Even, S., Cederbaum, I.: An algorithm for planarity testing of graphs. In: Rosenstiehl, P. (ed.) Theory of Graphs: International Symposium, pp. 215–232 (1967)Google Scholar
23. 23.
Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins University Press, Baltimore (2001)
24. 24.
Nishizeki, T., Chiba, N.: Planar graphs: theory and algorithms. Elsevier, Amsterdam (1988)
25. 25.
Nishizeki, T., Rahman, M.S.: Planar graph drawing. World Scientific, Singapore (2004)
26. 26.
Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17(4), 717–728 (2001)
27. 27.
Shih, W.K., Hsu, W.-L.: A new planarity test. Theoretical Computer Science 223(1-2), 179–191 (1999)

## Authors and Affiliations

• Bernhard Haeupler
• 1
• Krishnam Raju Jampani
• 2
• Anna Lubiw
• 2
1. 1.CSAIL, Dept. of Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA
2. 2.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada