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It seems to me that this is a special case of a graph connectivity situation. That is, the individual elements in foo can be seen as vertices, while foo[[i]] for i=1,2,3 can be seen as layers in a (directed) net. This can be visualized as: edges = {a -> c, a -> d, a -> e, a -> f, b -> c, b -> d, b -> e, b -> f, c -> x, d ...


I haven't yet tried to imagine an algorithm for your problem but here is a visualization that might help you approach it. pairs[ls_List] := Join @@ Tuples /@ Subsets[ls, {2}]; plot[ls_List, set_List] := ArrayPlot[ Outer[SubsetQ, set, pairs @ ls, 1] // Boole, FrameTicks -> All, FrameLabel -> {"Tuples", "Pairs"} ] If you are not using ...

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