Assuming $G$ is a planar graph with $k$ components, I need to determine an equation relating vertices, edges, faces and components. It is given that when $k=1$, $v-e+f=2$ (Euler's formula).
So from this I have gotten:
$v=e-f+2 \implies v=0-1+2=1$ (True)
$f=e-v+2 \implies f=0-1+2=1$ (True)
$e=v+f-2 \implies e=1+1-2=0$ (True)
From here I need to prove this. This is where I am getting confused. I can show multiple cases where these hold true, but I am having a hard time actually proving it.