Minimum path covering problem
It sounds like your graph is a transitively closed DAG, right? If so (and this is probably a restatement of what you say in your citation of Ntafos et al) the minimum number of paths needed to cover the DAG is just the maximum number of pairwise incomparable elements; this is Dilworth's theorem.

Your example may be simple enough that one can identify this maximum incomparable set directly, but in general it is possible to find this set in polynomial time, by an algorithm based in graph matching. The "Proof via König's theorem" section of the Wikipedia article on Dilworth's theorem explains how.