Minimal cuts are the generators of the cut ideal while the Alexander duality of path ideal generated by the minimal paths is the cut ideal -- more on Graph ideals here. Graph ideals are special case of Stanley-Reisner ideal that is explained in Sturmfels Combinatorial Commutative Algebra book.
Consider a cyclic graph such as $C_3$ or $C_4$. What are their minimal paths that specify their corresponding graph ideal?
What are the minimal cuts of $C_n$?
P.s. The computation of graph ideals and cut ideals here.