Years and Authors of Summarized Original Work
1976; Booth, Lueker
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
Auslander L, Parter SV (1961) On imbedding graphs in the plane. J Math Mech 10:517–523
Booth KS, Lueker GS (1976) Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J Comput Syst Sci 13:335–379
Boyer J, Myrvold W (1999) 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. Society for Industrial and Applied Mathematics, pp 140–146
Goldstein AJ (1963) An efficient and constructive algorithm for testing whether a graph can be embedded in the plane. In: Graph and combinatorics conference
Hopcroft J, Tarjan R (1974) Efficient planarity testing. J ACM 21:549–568
Lempel A, Even S, Cederbaum I (1967) An algorithm for planarity testing of graphs. In: Rosentiehl P (ed) Theory of graphs: international symposium. Gordon and Breach, New York, pp 215–232
Mehlhorn K, Mutzel P, Näher S (1993) An implementation of the hopcroft and tarjan planarity test. Technical report, MPI-I-93-151, Saarbrücken
Shih W-K, Hsu W-L (1999) A newplanarity test. Theor Comput Sci 223:179–191
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Borradaile, G. (2016). Planarity Testing. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_295
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