Planar Graphs · Challenge problem

Let $X=\{a,b,c\}$ and $Y=\{x,y,z\}$. Build a graph $G$ with all $6$ cross edges of $K_{3,3}$ EXCEPT $ax$, $by$, $cz$, together with $3$ new relay vertices $p,q,r$ and the paths $a\text{-}p\text{-}x$, $b\text{-}q\text{-}y$, $c\text{-}r\text{-}z$ in their place. By how much does the bipartite edge bound $2V-4$ EXCEED $G$'s actual edge count $E$?