Generally the theorem by Euler is stated:
If $G$ is connected and planar then $v - e + f = 2$ (where $v$ is the number of vertices, $e$ is the number of edges and $f$ is the number of faces of the graph $G$).
My question is:
Is this theorem an equivalence? I.e. is it true that if the equation $v - e + f = 2$ holds for a graph $G$ then $G$ is connected and planar.
Really thankful for any help with this. I couldn't find a counter-example myself, (i.e. a graph for which $v - e + f = 2$ holds but is either not connected or not planar) but I guess the theorem would be stated as an equivalence if it were an equivalence...
The fact that the graph is planar makes it clear what the faces are. If you take an abstract graph, you must specify where you will put the faces... and since you can put them everywhere you want, you can make the formula hold true for any graph you choose.