What values can $v-e+f$ attain if $G$ is a planar (non connected) graph? Let $G=(V,E)$ be a planar graph and choose planar representation.
If $G$ is connected, then according to Euler's formula, we have $$v − e + f = 2,$$ were $v$ is the number of vertices, $e$ the number of edges and $f$ the number of regions bounded by edges, including the outer, infinitely large region.
I was wondering which values $v-e+f$ can assume if $G$ is planar, but not connected. After some drawing, I feel that it can be any integer larges than 1, but I cannot prove it.
Is there someone who can help me out?

You can apply Euler's formula to each connected component. Beware of the exterior face!
You find: $\sum_i v_i - e_i + f_i = 2c$. And $\sum_i v_i = v, \sum_i e_i = e, \sum_i f_i = c + \sum_i (f_i - 1) = c + f - 1$.
Thus: $v - e + c + f - 1 = 2c$, which gives $v - e + f = c + 1$.
