I understand that the following is a misunderstanding of the definition of a planar graph, since the Euler characteristic of planar graphs being 2 is a long-standing theorem - I just can't see what it is!
What if we had the graph consisting of 2 points on the plane, with no edge between them. This is planar as it can easily be drawn on the plane. Now, V-E+F for this is 2-0+1=3 ... which is not 2...
What have I done wrong here?
The theorem states that any planar connected graph has $V-E+F = 2$. Note that if you have two planar connected graphs, their disjoint union embeds in the plane in the obvious way such that $V - E + F$ becomes $3$. Indeed, the "outside" faces of each of the two graphs now coincide so that instead of $2+2$ on the right-hand side (with each $2$ contributed by an individual connected component), we get $2+2-1$ (where the $-1$ removes the double-counted outside face).