Definition:Planar Graph/Face

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Definition

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

In the above, the faces are $BCEF$, $ABF$, $CFG$, $AFG$ and $ABCDCEG$.

Incident

Let $G = \struct {V, E}$ 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 $BCEF$ is incident to:

the edges $BC, CE, EF, FB$

the vertices $B, C, E, F$.

Adjacent

Let $G = \struct {V, E}$ 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, $BCEF$ and $ABF$ are adjacent, but $BCEF$ and $AFG$ are not adjacent.

Also see

• Results about faces of graphs can be found here.