How do I prove a graph is non planar?

Question

Using $E\le 3N - 6$

$$13\le 30$$

and using $E \le 2N - 4$

$$13\le 20$$

but I need to show that the graph is non planar

How can I?

Is it a subgraph of $K_{3,3}$?

Answer 1

Spot a subdivision of the $K3,3$ graph in your graph:

As $K3,3$ is not planar, your graph cannot be planar either.