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?

note that one of the faces of a planar graph is the unbounded region "outside" the graph. this is in common to the separate connected components, so you can then sum over components and obtain:
$$ v-e+f = 1 + C
$$
where $C$ is the number of connected components.
