(related to Lemma: Handshaking Lemma for Planar Graphs)

- By hypothesis, $G(V,E)$ is a biconnected planar graph with a given planar drawing $\mathcal D.$
- Since each edge in $\mathcal D$ has two sides, it contributes exactly $2$ to the sum of the face degrees (including the infinite face).
- Thus, $\sum_{f\in \mathcal D}\deg (f)=2\cdot|E|.$∎

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