planar graph

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Description

A planar graph is a graph which can be drawn on a plane (a flat 2-d surface) or on a sphere, with no edges crossing. When drawn on a sphere, the edges divide its area in a number of regions called faces (or “countries”, in the context of map coloring). When drawn on a plane, there is one outer country taking up all the space outside the drawing. Every graph drawn on a sphere can be drawn on a plane (puncture the sphere in the interior of any one of the countries) and vice versa. Statements on map coloring are often simpler in terms of a spherical map because the outer country is no longer a special case.

The number of faces (countries) equals $c+1$ where $c$ is the cyclomatic number, $c=m-n+k$ (where $m$ is the number of edges, $n$ the number of vertices, and $k$ the number of connected components of the graph). All this holds equally for planar multigraphs and pseudographs.

No complete graphs above $K_4$ are planar. $K_4$, drawn without crossings, looks like :

Hence it is planar (try this for $K_5$.)

A drawing of a planar graph is a drawing in which each edge is drawn as a line segment. Every planar graph has a drawing. This result was found independently by Wagner, Fáry and Stein. Schnyder improved this further by showing how to draw any planar graph with $n$ vertices on an integer grid of $O(n^2)$ area.

Definition

Ideally, this definition would just formalize what was described above. It will not, exactly. It will formalize the notion of a graph with an embedding into the plane.

Definition.

Let $M$ be a topological manifold. Then a graph on $M$ is a pair $(G,\iota )$, where

1. $G$ is a multigraph,

2. $\iota$ is a function from the graph topology of $G$ into $M$, and

3. $\iota$ is a homeomorphism onto its image.

A plane graph is a graph on ${ℝ}^2$.

A planar graph is a graph $G$ that has an embedding $\iota$ making $(G,\iota )$ into a plane graph.

Normally, the only manifolds $M$ that are of interest are two-dimensional.

The most usual question for which this definition finds a use is “can the following graph $G$ be made into a graph on $M$?”. When $M$ is the plane, this is usually phrased as “Is $G$ a planar graph?”. Wagner’s theorem provides a criterion for answering this question. When $M$ is a torus, the answer changes: the complete bipartite graph $K_{3,3}$ can be made into a graph on the torus.

A graph on a manifold has a notion of “face” as well as the usual graph notions of vertex and edge.

Title	planar graph
Canonical name	PlanarGraph
Date of creation	2013-03-22 12:17:37
Last modified on	2013-03-22 12:17:37
Owner	archibal (4430)
Last modified by	archibal (4430)
Numerical id	12
Author	archibal (4430)
Entry type	Definition
Classification	msc 05C10
Synonym	planar
Related topic	WagnersTheorem
Related topic	KuratowskisTheorem
Related topic	CrossingLemma
Related topic	CrossingNumber
Related topic	FourColorConjecture
Defines	plane graph