Encyclopedia of Algorithms

2008 Edition
| Editors: Ming-Yang Kao

Planarity Testing

1976; Booth, Lueker
  • Glencora Borradaile
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30162-4_295

Keywords and Synonyms

Planarity testing; Planar embedding      

Problem Definition

The problem is to determine whether or not the input graph G is planar. The definition pertinent to planarity-testing algorithms is: G is planar if there is an embedding of G into the plane (vertices of G are mapped to distinct points and edges of G are mapped to curves between their respective endpoints) such that edges do not cross. Algorithms that test the planarity of a graph can be modified to obtain such an embedding of the graph.

Key Results

Theorem 1

There is an algorithm that given a graph G with n vertices, determines whether or not G is planar in O(n) time.

The first linear-time algorithm was obtained by Hopcroft and Tarjan [5] by analyzing an iterative version of a recursive algorithm suggested by Auslander and Parter [1] and corrected by Goldstein [4]. The algorithm is based on the observation that a connected graph is planar if and only if all its biconnected components are planar. The...

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Recommended Reading

  1. 1.
    Auslander, L., Parter, S.V.: On imbedding graphs in the plane. J. Math. and Mech. 10, pp. 517–523 (1961)Google Scholar
  2. 2.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comp. Syst. Sci. 13, pp. 335–379 (1976)Google Scholar
  3. 3.
    Boyer, J., Myrvold, W.: Stop minding your P's and Q's: A simplified O(n) planar embedding algorithm. In: SODA '99: Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia, PA, USA, Society for Industrial and Applied Mathematics, pp. 140–146 (1999)Google Scholar
  4. 4.
    Goldstein, A.J.: An efficient and constructive algorithm for testing whether a graph can be embedded in the plane. In: Graph and Combinatorics Conf. (1963)Google Scholar
  5. 5.
    Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. ACM 21, pp. 549–568 (1974)Google Scholar
  6. 6.
    Lempel, A., Even, S., Cederbaum, I.: An algorithm for planarity testing of graphs. In: Rosentiehl, P. (ed.) Theory of Graphs: International Symposium. New York, Gordon and Breach, pp. 215–232 (1967)Google Scholar
  7. 7.
    Mehlhorn, K., Mutzel, P., Näher, S.: An implementation of the hopcroft and tarjan planarity test. Tech. Rep. MPI-I-93-151, Saarbrücken (1993)Google Scholar
  8. 8.
    Shih, W.-K., Hsu, W.-L.: A new planarity test. Theor. Comput. Sci. 223, pp. 179–191 (1999)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Glencora Borradaile
    • 1
  1. 1.Department of Computer ScienceBrown UniversityProvidenceUSA