# Definition:Planar Graph

## Definition

A **planar graph** is a graph which can be drawn in the plane (e.g. on a piece of paper) without any of the edges crossing over, that is, meeting at points other than the vertices.

This is a **planar graph**:

### Face

The **faces** of a planar graph are the areas which are surrounded by edges.

In the above, the **faces** are $ABHC$, $CEGH$, $ACD$, $CDFE$ and $ADFEGHIHB$.

### Incident

Let $G = \left({V, E}\right)$ be a planar graph:

Then a face of $G$ is **incident to** an edge $e$ of $G$ if $e$ is one of those which surrounds the face.

Similarly, a face of $G$ is **incident to** a vertex $v$ of $G$ if $v$ is at the end of one of those incident edges.

In the above graph, for example, the face $ABHC$ is incident to:

### Adjacent

Let $G = \left({V, E}\right)$ be a planar graph.

Two faces of $G$ are **adjacent** if and only if they are both incident to the same edge (or edges).

In the above diagram, $ABHC$ and $ACD$ are adjacent, but $ABHC$ and $CDFE$ are *not* adjacent.

## Non-Planar

A **non-planar graph** is a graph which is not planar.

This is a **non-planar graph**: