The argument we have outlined above is not quite correct, since we made the unjustified assumption that all graphs can be built up from \(P_2\) using only the two moves we described. To avoid this issue, we can use a minimal criminal argument. You are asked to do this in the exercises, but the idea is essentially the same as we have here, except that we start with a minimal connected planar graph that does not satisfy the formula, then remove either an edge or a vertex (and its edge) to get a smaller connected planar graph that does satisfy the formula. But just like the adding moves we have described above, removing an edge or a vertex does not change the quantity \(v - e + f\text{.}\)
