Bipartite Graphs and Matchings · Hard problem

A network splits into two separate pieces. In the first, people $m_1,m_2$ can each be assigned only to resource $s$. In the second, items $a,b,c,d,e$ are linked in a path $a\text{-}b\text{-}c\text{-}d\text{-}e$. Treating every person, resource, and item together as $8$ vertices joined by these links, what is the largest number of them that can be chosen so that no chosen vertex is linked to another chosen vertex?