Euler's formula for non-planar graph

Question

Could anyone please provide an example of non-planar graph, that corresponds to Euler's formula: $$v - e + f = 2$$

or show that there is no such a graph?

Answer 1

In general, Euler's formula can only be used to prove that a graph is non-planar. To prove planarity, one usually either finds an actual plane drawing or shows that the graph contains no $K_5$ or $K_{3,3}$ minors.