Definition:Planar Graph
Definition
A planar graph is a graph which can be drawn in the plane (for example, on a piece of paper) without any of the edges crossing over, that is, meeting at points other than the vertices.
Examples
Arbitrary Example
This is a planar graph:
Complete Bipartite Graph K2,3
The complete bipartite graph $K_{2, 3}$ 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 $BCEF$, $ABF$, $CFG$, $AFG$ 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:
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.
Non-Planar
A non-planar graph is a graph which is not planar.
This is a non-planar graph:
Also see
- Results about planar graphs can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): graph: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): graph: 2.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Euler's Theorem
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): planar graph