# The Complexity of Planarity Testing

Conference paper

First Online:

## Abstract

We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for **L**, and lies in **SL**. This nearly settles the question, since it is widely conjectured that **L** = **SL** [25]. The upper bound of **SL** matches the lower bound of **L** in the context of (nonuniform) circuit complexity, since **L**/poly is equal to **SL**/poly.

Similarly, we show that a planar embedding, when one exists, can be found in **FL** ^{SL}.

Previously, these problems were known to reside in the complexity class **AC** ^{1}, via a *O*(log *n*) time CRCW PRAM algorithm [22], although planarity checking for degree-three graphs had been shown to be in **SL** [23, 20].

## Keywords

Span Tree Planarity Testing Constraint Graph Fundamental Cycle Euler Tour
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lovász, and C. Rackoff. Random walks, universal traversal sequences, and the complexity of maze problems. In
*Proceedings of the 20th Annual Symposium on Foundations of Computer Science*, pages 218–223. IEEE, 1979.Google Scholar - [2]C. Àlvarez and R. Greenlaw. A compendium of problems complete for symmetric logarithmic space. Technical Report ECCC-TR96-039, Electronic Colloquium on Computational Complexity, 1996.Google Scholar
- [3]R. Armoni, A. Ta-Shma, A. Wigderson, and S. Zhou. \( SL \subseteq L^{\tfrac{4} {3}} \). In
*Proceedings of the 29th Annual Symposium on Theory of Computing*, pages 230–239. ACM, 1997.Google Scholar - [4]K. Booth and G. Lueker. 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.zbMATHMathSciNetGoogle Scholar - [5]A. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility.
*SIAM Journal on Computing*, 13(2):423–439, 1984.zbMATHCrossRefMathSciNetGoogle Scholar - [6]S. A. Cook and P. McKenzie. Problems complete for L.
*Journal of Algorithms*, 8:385–394, 1987.zbMATHCrossRefMathSciNetGoogle Scholar - [7]K. Etessami. Counting quantifiers, successor relations, and logarithmic space.
*Journal of Computer and System Sciences*, 54(3):400–411, Jun 1997.zbMATHCrossRefMathSciNetGoogle Scholar - [8]S. Even and R. Tarjan. Computing an
*st*-numbering.*Theoretical Computer Science*, 2:339–344, 1976.zbMATHCrossRefMathSciNetGoogle Scholar - [9]A. Gál and A. Wigderson. Boolean vs. arithmetic complexity classes: randomized reductions.
*Random Structures and Algorithms*, 9:99–111, 1996.zbMATHCrossRefMathSciNetGoogle Scholar - [10]J. Hopcroft and R. Tarjan. Efficient planarity testing.
*Journal of the ACM*, 21:549–568, 1974.zbMATHCrossRefMathSciNetGoogle Scholar - [11]N. Immerman. Nondeterministic space is closed under complementation.
*SIAM Journal on Computing*, 17(5):935–938, Oct 1988.zbMATHCrossRefMathSciNetGoogle Scholar - [12]J. Ja’Ja’ and J. Simon. Parallel algorithms in graph theory: Planarity testing.
*SIAM Journal on Computing*, 11:314–328, 1982.CrossRefMathSciNetGoogle Scholar - [13]J. Ja’Ja’ and J. Simon. Space effcient algorithms for some graph-theoretic problems.
*Acta Informatica*, 17:411–423, 1982.CrossRefMathSciNetGoogle Scholar - [14]M. Karchmer and A. Wigderson. On span programs. In
*Proceedings of the 8th Conference on Structure in Complexity Theory*, pages 102–111. IEEE Computer Society Press, 1993.Google Scholar - [15]R. M. Karp and R. J. Lipton. Turing machines that take advice.
*L’ Ensignement Mathématique*, 28:191–210, 1982.zbMATHMathSciNetGoogle Scholar - [16]A. Lempel, S. Even, and I. Cederbaum. An algorithm for planarity testing in graphs. In
*Theory of Graphs: International Symposium*, pages 215–232, New York, 1967. Gordon and Breach.Google Scholar - [17]M. Mahajan, P. R. Subramanya, and V. Vinay. A combinatorial algorithm for Pfaffians. In
*Proceedings of the Fifth Annual International Computing and Combinatorics Conference COCOON*,*LNCS Volume 1627*, pages 134–143. Springer-Verlag, 1999. DIMACS Technical Report 99-39.Google Scholar - [18]Y. Maon, B. Schieber, and U. Vishkin. Parallel ear decomposition search (EDS) and
*st*-numbering in graphs.*Theoretical Computer Science*, 47:277–296, 1986.zbMATHCrossRefMathSciNetGoogle Scholar - [19]N. Nisan, E. Szemeredi, and A. Wigderson. Undirected connectivity in
*O*(log^{1.5}*n*) space. In*Proceedings of the 33rd Annual Smposium on Foundations of Computer Science*, pages 24–29. IEEE Computer Society Press, 1992.Google Scholar - [20]N. Nisan and A. Ta-Shma. Symmetric Logspace is closed under complement.
*Chicago Journal of Theoretical Computer Science*, 1995.Google Scholar - [21]V. Ramachandran. Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. In J. Reif, editor,
*Synthesis of Parallel Algorithms*. Morgan Kaumann, 1993.Google Scholar - [22]V. Ramachandran and J. Reif. Planarity testing in parallel.
*Journal of Computer and System Sciences*, 49:517–561, 1994.zbMATHCrossRefMathSciNetGoogle Scholar - [23]J. Reif. Symmetric complementation.
*Journal of the ACM*, 31(2):401–421, 1984.zbMATHCrossRefMathSciNetGoogle Scholar - [24]K. Reinhardt and E. Allender. Making nondeterminism unambiguous. In
*38 th IEEE Symposium on Foundations of Computer Science (FOCS)*, pages 244–253, 1997. to appear in SIAM J. Comput.Google Scholar - [25]M. Saks. Randomization and derandomization in space-bounded computation. In
*Proceedings of the 11th Annual Conference on Computational Complexity*, pages 128–149. IEEE Computer Society, 1996.Google Scholar - [26]W. J. Savitch. Relationships between nondeterministic and deterministic tape complexities.
*Journal of Computer and System Sciences*, 4(2):177–192, April 1970.zbMATHMathSciNetGoogle Scholar - [27]R. Szelepcsényi. The method of forced enumeration for nondeterministic automata.
*Acta Informatica*, 26(3):279–284, 1988.zbMATHCrossRefMathSciNetGoogle Scholar - [28]W. T. Tutte. Toward a theory of crossing numbers.
*Journal of Combinatorial Theory*, 8:45–53, 1970.zbMATHCrossRefMathSciNetGoogle Scholar - [29]H. Whitney. Non-separable and planar graphs.
*Transactions of the American Mathematical Society*, 34:339–362, 1932.zbMATHCrossRefMathSciNetGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2000