Proving a graph not simple and planar after adding edges

[cmkluza]

Homework Statement

Prove that no more than 3 edges can be added to G while keeping it planar and simple.
(The graph in the image is graph G)

Homework Equations

For a connected, planar, simple graph G, with E edges and V vertices:
E ≤ 3V - 6

The Attempt at a Solution

At first, we have 9 edges, and 6 vertices:

9 ≤ 3(6) - 6 → 9 ≤ 12

However, if I just add 4 vertices off to the side, say to the right, and draw four edges connecting them all to vertex E, we'd have 13 edges and 10 vertices.

13 ≤ 3(10) - 6 → 13 ≤ 24 which is still true.

Am I misunderstanding one of the definitions in this problem? I'm not quite getting how what I'm doing isn't keeping the graph planar and simple.

Edit: Clarified equation

 

[haruspex]

I'm not quite getting how what I'm doing isn't keeping the graph planar and simple.

It does not say you can add any vertices.

 

[cmkluza]

It does not say you can add any vertices.

Ah, ok, not sure where my assumption that I could came from. So, does it suffice as mathematical proof to just show that:

9 ≤ 12
9 + 3 ≤ 12 - is true
9 + 4 ≤ 12 - is not true

Thanks!

 

[haruspex]

Ah, ok, not sure where my assumption that I could came from. So, does it suffice as mathematical proof to just show that:

9 ≤ 12
9 + 3 ≤ 12 - is true
9 + 4 ≤ 12 - is not true

Thanks!

No. The inequality is necessary but not sufficient. Consider K_3,3. It has 9 edges, 6 vertices, but is not planar.

 

[epenguin]

So, any ideas or plan?
I think the 6 vertices are inviting you to try and show the result of adding more than 3 edges would contain K_3,3 which is non-planar.