Euler's formula for connected, planar graphs.

Question

So I have to prove the graph in the picture is planar using Euler's formula $v - e + f = 2.$ It obviously is planar, but the formula for it is $9 - 12 + 6 = 3$. What am I doing wrong?

Answer 1

1. I have to prove the graph in the picture is planar using Euler's formula

You can't prove planarity using Euler's formula. You can sometimes prove the opposite: that graph is not planar, by using Euler's formula and fact that each face in planar graph has at least 3 edges. (which will get your criteria that in planar graph it should be that $E \leq 3V - 6$).

2. Euler's formula is for connected graphs. If your graph is disconnected, calculate Euler's formula for each connected part separately. Alternatively you can use formula $V - E + F = k + 1$, where k $-$ number of connected components.