(related to Corollary: Number of Vertices, Edges, and Faces of a Dual Graph)

- By hypothesis, $G$ is a connected planar graph with a planar drawing $\mathcal D,$ and with $|V|$ vertices, $|E|$ edges and $|F|$ faces.
- It follows from the construction of the dual graph $G^*_{\mathcal D}$ that $G^*_{\mathcal D}$ has $|F|$ vertices and $|E|$ edges and that it is a planar graph.
- From the Euler's formula for planar graphs he have:
- For $G$: $|V|-|E|+|F|=2.$
- For $G^*_{\mathcal D}$: $|F|-|E|+f=2,$ where $f$ is the (unknown) number of face of $G^*_{\mathcal D}.$

- Comparing both equations, we get $f=|V|.$∎

**Aldous Joan M., Wilson Robin J.**: "Graphs and Applications - An Introductory Approach", Springer, 2000