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:
