Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / R / 1 / ii / b / Solution

The modified graph is planar. Place $x_1=(-2,0)$ and $x_2=(2,0)$, and place the other vertices on the vertical line between them in the bottom-to-top order $y_2,y_1,y_5,y_3,y_4$. Draw each edge $x_i{y}_j$ as a straight segment. Each half of the drawing is a fan sharing one endpoint, so its edges do not cross. The added edges $y_2{y}_1$ and $y_1{y}_5$ are consecutive vertical segments, meeting other edges only at their endpoints. The following original drawing exhibits the embedding.