According to wikipedia, the Euler characteristic of planar graph is $2$. I'm confused about why this would be true, since for any simple polygon drawn with $n$ vertices we have $e=v=n$ and $f=1$, so $\chi=n-n+1=1$. And of course thinking of the Euler characteristic as a homotopy invariant, the Euler characteristic of a disc is $1$.
It almost seems as if when calculating the Euler characteristic of a planar graph they are counting the empty space around the graph as an additional face, as if that graph were drawn on the surface of a sphere instead of in the plane.
Is there a reason for this seemingly strange convention with ''planar'' graphs?