IR-TST 2002 Problem 8

We call $A_{1},A_{2},A_{3}$ mangool iff there is a permutation $\pi$ that $A_{\pi(2)}\not\subset A_{\pi(1)},A_{\pi(3)}\not\subset A_{\pi(1)}\cup A_{\pi(2)}$ . A good family is a family of finite subsets of $\mathbb N$ like $X,A_{1},A_{2},\dots,A_{n}$ . To each goo family we correspond a graph with vertices $\{A_{1},A_{2},\dots,A_{n}\}$ . Connect $A_{i},A_{j}$ iff $X,A_{i},A_{j}$ are mangool sets. Find all graphs that we can find a good family corresponding to it.