# Definition: Planar Drawing (Embedding)

Let $$\mathbb R^2$$ be an Euclidean plane and let $$a,b\in\mathbb R$$, $$b > a$$ be two real numbers. Furthermore, let $$I=[a,b]$$ be a closed real interval. . A planar drawing (or a planar embedding) $$\mathcal D$$ of an undirected graph $$G(V,E,\gamma)$$ consists of

$$(i)$$ an injective function1 $$f:V\to \mathbb R^2,$$

$$(ii)$$ for each edge $$e_i\in E$$, functions $$\epsilon_i: I\to \mathbb R^2$$ such that: * $$\epsilon_i$$ is a simple closed curve, if $$e_i$$ is a loop2, * $$\epsilon_i$$ is a simple curve, if $$e_i$$ is not a loop3, * if $$x,y$$ are the ends of the edge $$e_i$$, we either have $$\epsilon_i(a)=f(x)$$ and $$\epsilon_i(b)=f(y)$$ or we have $$\epsilon_i(a)=f(y)$$ and $$\epsilon_i(b)=f(x)$$, depending on at which vertex the curve starts and at which it ends4. * The images of the restrictions of the curves $$e_i$$ on the open interval $$J:=(a,b)$$, i.e. the functions $${\epsilon_i|}_{J} : I \to \mathbb R^2$$ are mutually exclusive5, formally $\epsilon_i(J)\cap\epsilon_j(J)=\emptyset\quad\quad i\neq j.$

### Example

A graph:

and examples of its planar drawings:

Please note that a graph can have different planar drawings.

For educational reasons, each vertex has been drawn not as a point in the plane, but as a labeled circle to enable the reader to compare all graphs.

Chapters: 1
Corollaries: 2
Definitions: 3 4 5 6 7
Lemmas: 8 9
Proofs: 10 11 12 13 14 15 16
Theorems: 17 18

Thank you to the contributors under CC BY-SA 4.0!

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### References

#### Bibliography

1. Matoušek, J; Nešetşil, J: "Invitation to Discrete Mathematics", Oxford University Press, 1998

#### Footnotes

1. The injectivity of $$f$$ makes sure that $$f$$ assigns to each vertex of $$G$$ a unique point in the Euclidean plane $$\mathbb R^2$$, i.e. the same point of the plane is never assigned to two different vertices.

2. This property ensures that the drawing $$\epsilon_i$$ of a loop does not cross itself.

3. This property ensures that the drawing $$\epsilon_i$$ of the edge does not cross itself.

4. Please note that since $$f(x), f(y)$$ are the points in the plane, which are drawings of the vertices $$x,y$$, this property ensures that the drawing of any edge connecting these vertices (i.e. the curve $$\epsilon_i$$) starts ($$\epsilon_i(a)$$) and ends ($$\epsilon_i(b)$$) exactly at these points in the plane.

5. This property makes sure that the drawings of any two edges do not cross, except at the endpoints (i.e. the vertices).
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