# Planar graph ¶

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A planar graph (Graphe planaire), is a graph that could be drawn without having two edges crossing each other. We are calling planar representation/Graphe planaire topologique, the planar representation of a planar graph.

When we are drawing a planar graph, we will have

## Face ¶

Do you know the bucket tool in image editors? A face is something colored by the bucked tool in one go. You got at least one unbounded face, which is the one not colored after coloring all of your faces.

A face is a surface of the graph delimited by edges. It can be either a

• bounded face/face finie
• external, outer or unbounded face/face infinie

## Dual graph ¶

We are calling Dual graph (Graphe dual), the graph made using the faces as vertices. If we have two faces $F_1$ and $F_2,$ and if $A \in F_1$ was adjacent to $B \in F_2$ then we have $F_1$ is adjacent to $F_2$.

## Bounds ¶

Note that you can only use a vertex once inside the bounds (Frontière/contour) of a face. The bounds are a list of edges you used to delimit a face. The bounds are forming a cycle basis in a planar representation.

## Graph minor ¶

You can get the minor of a graph by

• removing an edge
• contracting an edge (the vertex and their edges are merged)
• deleting an edge having a degree of one (isolated, $d(vertex) = 1$)

## Check if the graph is planar ¶

Let $m=\#edges=|E|$, $n=\#vertices=|V|$ and $f=\#faces$, then a graph is planar if

• Euler's formula: $n-m+f=2$ (you can find $f$ with $f=m-n+2$)
• if $m < 3n - 5$ in a connected graph
• Kuratowski's theorem
• The graphs $K_{3,3}$ and $K_5$ are not planar
• A subgraph of these is not planar
• Robertson–Seymour theorem
• A graph minor of $K_{3,3}$ and $K_5$ is not planar

Other notes

• if the $\forall{i \in V},\ d(i) \gt 5$, the graph is NOT planar
• if a graph is planar, then $\exists{i \in V},\ d(i) \lt 6$
• According to the English wikipedia, a graph is planar if
• $m \le 3n - 6$ (yeah it's 6 not 5)
• $f \le 2n - 4$
• if there are no cycles of length 3, $m \le 2n-4$

## Example 1 ¶

Are the following graphs planar?

The first graph is planar because it's $K_5$.

The second one is planar, because I found a planar representation. Notice that we have $n + m = f = 2$ ($n=5$, $m=9$, $f=6$=five+the outer face)

The third graph has a subgraph $K_{3,3}$ so it's not planar.

The fourth graph $m=12 \le 6 * 3 - 5 \le 13$ so the graph is planar. Simply move the vertices "4" and "5".

All degrees are equals to $6$, so the graph is not planar. We could have used $m < 3n - 5$.

As for the sixth graph, it's not planar either. I used Robertson–Seymour theorem and found that the minor of the graph (by only merging vertices) is $K_6$ so if I remove one more vertex, it's $K_5$ and $K_5$ is not a planar graph.

## Example 2 ¶

Use Robertson–Seymour theorem and demonstrate that the Petersen graph is not planar.

This answer can be found on Wikipedia source. Here is a copy