Bipartite Graphs and Matchings · Hard problem

A network of $12$ routers is built from three squares chained together: $1\text{-}2\text{-}3\text{-}4\text{-}1$, then $4\text{-}5\text{-}6\text{-}7\text{-}4$ (sharing router $4$ with the first), then $7\text{-}8\text{-}9\text{-}10\text{-}7$ (sharing router $7$ with the second), plus two extra routers $11$ and $12$ each linked only to router $1$. Every connected bipartite graph splits into two groups with every link crossing between them. What is the size of the LARGER group?