It's a famous fact that for all connected, planar graphs, Euler observed (but failed, remarkably, to prove) that $$V-E+F=2$$ However, all of the proofs I have seen (including those I have created myself) are one-way, i.e. they show that if a graph satisfies the above conditions then the polyhedral formula holds (typically because the proof involves gradually deconstructing one's graph in a manner that leaves $V-E+F$ invariant). Does the converse in general hold?
That is, if $V-E+F =2$ for some graph, is that enough to conclude that it's planar and connected?