# Encyclopedia of Algorithms

2008 Edition
| Editors: Ming-Yang Kao

# Planarity Testing

1976; Booth, Lueker
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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