Bipartite Graphs and Matchings · Superchallenge problem

A network has three separate pieces with no links between them: a $4$-cycle $1\text{-}2\text{-}3\text{-}4\text{-}1$, a $6$-cycle $5\text{-}6\text{-}7\text{-}8\text{-}9\text{-}10\text{-}5$, and a piece where $u_1,u_2$ are each linkable to any of $p,q,r$. Treating everyone as $15$ vertices, what is the largest number of them that can be chosen so that no chosen vertex is linked to another chosen vertex?