 Solution

A dependency graph of events, written here as a loopless directed graph $D$ on the event indices, must satisfy a joint independence condition: for each $i$, $A_i$ is independent of the sigma-algebra generated by all $A_j$ with $j\ne i$ and no arc $i\to j$. In particular this means independence from every Boolean combination of those non-neighbour events, not merely pairwise independence. The outgoing neighbours list the events for which that independence is not required.