$G$ is a planar, connected graph. Let $n$ be number of vertices, $m$ number of edges and $f$ number of faces. Prove that dual graph $G*$ has $f(G*)=n$

Question

I understand how planar graphs and their dual graphs are connected, but I can't find a way to prove this because it just seems..obvious?

Answer 1

*sorry in advance if I mess something up, I am a bit rusty with graph theory.

For this I'm going to use Euler's formula for graphs, where the number of vertices plus the number of faces, minus the number of edges equals two for planar graphs ($n+f-m=2$)

Using Euler's formula, we set up the following equation (n* denotes the vertices of G* and so on): $$n*+f*-m*=2=2=n+f-m$$

Because the number of edges of a planar graph equals the number of edges of its corresponding dual graph (when constructing the dual graph there is a one to one correspondence), we can reduce the formula to: $$n*+f*=n+f$$ By definition, the number of faces of graph G should be equal to the number of vertices of G* (n*=f). So we can cross those out. $$f*=n$$ which yields the desired result.