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I have from my notes that they claim in a planar graph,
$$2|E| = \text{sum of all degrees of regions}$$
where $|E|$ is the cardinality of the edge-set of the graph
They say this because each edge in the graph contributes twice to the degree of the region.
However, I can't see this in the following example:
enter image description here

The inner region seems to have degree $4$, however the outer region seems to have degree $5$. (from edges AB,BC,CD,DE,EB)

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For this identity to hold, we need to count the edge $AB$ twice, on the basis that as you go around the boundary of the outer face, you trace that edge twice: once going from $A$ to $B$ and once from $B$ to $A$.

enter image description here

If we follow this convention, then the inner region (blue) has degree $4$ while the outer region (red) has degree $6$, and everything works out.

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