Bipartite Graphs and Matchings · Hard problem

A $10$-item network has $X=\{1,2,3,4,5\}$ and $Y=\{a,b,c,d,e\}$: items $1,2,3$ are each compatible only with $a$ or $b$; item $4$ is compatible with $c$ or $d$; item $5$ is compatible with $d$ or $e$. What is the size of a maximum independent set among all $10$ items (the largest group with no compatibility link between any two of its own members)?