Abstract
The data structure SPQR-tree represents the decomposition of a biconnected graph with respect to its triconnected components. SPQR-trees have been introduced by Di Battista and Tamassia [8] and, since then, became quite important in the field of graph algorithms. Theoretical papers using SPQR-trees claim that they can be implemented in linear time using a modification of the algorithm by Hopcroft and Tarjan [15] for decomposing a graph into its triconnected components. So far no correct linear time implementation of either triconnectivity decomposition or SPQR-trees is known to us. Here, we show the incorrectness of the Hopcroft and Tarjan algorithm [15], and correct the faulty parts. We describe the relationship between SPQR-trees and triconnected components and apply the resulting algorithm to the computation of SPQR-trees. Our implementation is publically available in AGD [1].
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Partially supported by DFG-Grant Mu 1129/3-1, Forschungsschwerpunkt “Effiziente Algorithmen für diskrete Probleme und ihre Anwendungen”.
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References
G. Di Battista and R. Tamassia. On-line graph algorithms with SPQR-trees. In M. S. Paterson, editor, Proc. of the 17th International Colloqium on Automata, Languages and Prog ramming (ICALP), volume 443 of Lecture Notes in Computer Science, pages 598–611. Springer-Verlag, 1990.
P. Bertolazzi, G. Di Battista, and W. Didimo. Computing orthogonal drawings with the minimum number of bends. In Proc. 5th Workshop Algorithms, Data Struct., volume 1272 of Lecture Notes in Computer Science, pages 331–344, 1998.
P. Bertolazzi, G. Di Battista, G. Liotta, and C. Mannino. Optimal upward planarity testing of single-source digraphs. SIAM J. Comput., 27(1):132–169, 1998.
D. Bienstock and C. L. Monma. On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica, 5(1):93–109, 1990.
Elias Dahlhaus. A linear time algorithm to recognize clustered planar graphs and its parallelization. In C. L. Lucchesi and A. V. Moura, editors, LATIN’ 98: Theoretical Informatics, Third Latin American Symposium, volume 1380 of Lecture Notes in Computer Science, pages 239–248. Springer-Verlag, 1998.
G. Di Battista, A. Garg, G. Liotta, R. Tamassia, and F. Vargiu. An experimental comparision of four graph drawing algorithms. Comput. Geom. Theory Appl., 7:303–326, 1997.
G. Di Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th IEEE Symp. on Foundations of Computer Science, pages 436–441, 1989.
G. Di Battista and R. Tamassia. On-line maintanance of triconnected components with SPQR-trees. Algorithmica, 15:302–318, 1996.
G. Di Battista and R. Tamassia. On-line planarity testing. SIAM J. Comput., 25(5):956–997, 1996.
W. Didimo. Dipartimento di Informatica e Automazione, Università di Roma Tre, Rome, Italy, Personal Communication.
C. Gutwenger and P. Mutzel. A linear time implementation of SPQR-trees. Technical report, Technische Universität Wien, 2000. To appear.
C. Gutwenger, P. Mutzel, and R. Weiskircher. Inserting an edge into a planar graph. In Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’ 01). ACM Press, 2001. To appear.
J. E. Hopcroft and R. E. Tarjan. Dividing a graph into triconnected components. SIAM J. Comput., 2(3):135–158, 1973.
J. E. Hopcroft and R. E. Tarjan. Efficient planarity testing. Journal of the ACM, 21:549–568, 1974.
G. Kant. Drawing planar graphs using the canonical ordering. Algorithmica, Special Issue on Graph Drawing, 16(1):4–32, 1996.
T. Lengauer. Hierarchical planarity testing. Journal of the ACM, 36:474–509, 1989.
S. MacLaine. A structural characterization of planar combinatorial graphs. Duke Math. J., 3:460–472, 1937.
K. Mehlhorn and S. Näher. The LEDA Platform of Combinatorial and Geometric Computing. Cambridge University Press, 1999. to appear.
P. Mutzel and R. Weiskircher. Optimizing over all combinatorial embeddings of a planar graph. In G. Cornuéjols, R. Burkard, and G. Woeginger, editors, Proceedings of the Seventh Conference on Integer Programming and Combinatorial Optimization (IPCO), volume 1610 of LNCS, pages 361–376. Springer Verlag, 1999.
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Gutwenger, C., Mutzel, P. (2001). A Linear Time Implementation of SPQR-Trees. In: Marks, J. (eds) Graph Drawing. GD 2000. Lecture Notes in Computer Science, vol 1984. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44541-2_8
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DOI: https://doi.org/10.1007/3-540-44541-2_8
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