planar graph

mathematics

Written by

The Information Architects of Encyclopaedia Britannica

Last updated

July 1, 2026

Quick Summary

A planar graph is a graph that can be drawn on a flat surface, such as a piece of paper, without any of its lines crossing each other except at the points where they connect. In graph theory, a graph consists of points (vertices) connected by lines (edges).12

Planar graphs are a key concept in topological graph theory. A famous theorem by Kazimierz Kuratowski states that a graph is planar if and only if it does not contain a specific type of subgraph that is related to either the complete graph on five vertices (K₅) or the complete bipartite graph K_3,3. This concept has applications in fields like chemistry, where it can relate to the structure of molecules.21

Summary created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

2 Britannica Sources

Ask Britannica AI about this topic

Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.

combinatorics
Planar graphs
A graph G is said to be planar if it can be represented on a plane in such a fashion that the vertices are all distinct points, the edges are simple curves, and no two edges meet one another except at their terminals.…
Read More
A graph G is said to be planar if it can be represented on a plane in such a fashion that the vertices are all distinct points, the edges are simple curves, and no two edges meet one another except at their terminals.…
Read More

number game
Graphs and networks
A planar graph is one in which the edges have no intersection or common points except at the edges. (It should be noted that the edges of a graph need not be straight lines.) Thus a nonplanar graph can be transformed into an equivalent, or isomorphic,…
Read More
A planar graph is one in which the edges have no intersection or common points except at the edges. (It should be noted that the edges of a graph need not be straight lines.) Thus a nonplanar graph can be transformed into an equivalent, or isomorphic,…
Read More

graph theory
…problem in this area concerns planar graphs. These are graphs that can be drawn as dot-and-line diagrams on a plane (or, equivalently, on a sphere) without any edges crossing except at the vertices where they meet. Complete graphs with four or fewer vertices are planar, but complete graphs with five…
Read More
…problem in this area concerns planar graphs. These are graphs that can be drawn as dot-and-line diagrams on a plane (or, equivalently, on a sphere) without any edges crossing except at the vertices where they meet. Complete graphs with four or fewer vertices are planar, but complete graphs with five…
Read More