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Should every node's value be printed out exactly once and if so, are there nodes which have two parents?

If so, I would keep the pointers in a set to mark nodes as processed but is there a more elegant solution?

I would explicitly tell them that you should revisit nodes, and give them a few examples illustrating that point -- so all they needed to do was traverse edges.

Each node is an object and has a "checked" variable.

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