Euler's formula about graphs embedded in $\mathbb{R^2}$

Question

State and prove Euler's formula about graphs embedded into $\mathbb{R^2}$

I know that if we suppose $ G $ is a finite connected graph drawn on the surface of a sphere $ S^2 $. Then the complement $ S^2 -G $ consists of $ F=2-\chi(G) $ `faces,' connected regions homeomorphic to open discs.

But is the statement for $\mathbb{R^2}$?

Answer 1

I think what you mean is Euler's polyhedral formula. It states that for every convex polyhedron with $V$ vertices (corners), $E$ edges and $F$ faces the formula $V-E+F=2$ holds.

The same statement holds for planar graphs. So the statement for $\mathbb R^2$ is that the formula holds if your graph is a planar graph.