---
tag: 029JQEK15B
kind: definition
name: Planar graph
msc: [05C10]
defines: 0NC9R64TXP
sources:
  - url: https://en.wikipedia.org/wiki/Planar_graph
    locator: lead section
---

## Definition

A [[graph|0JD5C82QMB]] $G$ is planar if it has a drawing in the plane in which no two edges cross.
In such a drawing, the vertices are distinct points of $\mathbb{R}^2$, and each edge $uv$ is a simple arc from $u$ to $v$.
Two arcs meet only at a common end, and an arc contains no vertex other than its ends.

## Remarks

A drawing of this kind is a plane graph, or an embedding of $G$ in the plane.
The complete graph $K_4$ is planar. [[$K_5$ and $K_{3,3}$ are not planar|030A3FHN14]].
[[Kuratowski's theorem|0J6G5NJC5R]] characterizes the finite planar graphs by these two graphs.

## Explanations

### L1 human

A graph is planar when you can draw it on paper with no crossing edges.
A map of countries gives a planar graph: one vertex for each country, and an edge between neighbours.