The converse of the Upper Bound Theorem would state that a graph which satisfies the inequality $\displaystyle e \leq { \frac{n (v-2)}{n-2} $ is planar.

This converse is not true as seen in picture.

Verify that the inequality $\displaystyle e \leq { \frac{n (v-2)}{n-2} $ is true for this graph. Once done, use the inside-outside algorithm to show that the graph is actually non-planar.