Exercise 14.

Give a careful proof that the graph below is not planar.

A graph with 11 vertices. A single vertex in the center, then five vertices equally spaced around a ring around it, and five more equally spaced around a ring around those. Edges form the sides of a pentagon for the outer ring of vertices. Each outer vertex is also adjacent to two inner vertices: the two on either side of the vertex closest to it. Finally, every inner vertex is also adjacent to the center vertex.
Hint.
in-context