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Graph ideals are a special case of Stanley-Reisner ideal, explained in Combinatorial Commutative Algebra book by Sturmfels, and graph ideals here. Graph ideals are generated by the minimal paths while cut ideals are generated by minimal cuts. Computing them may be done somehow with Gröbner basis and Buchberger's algorithm.

Graph ideals are in the square-free monomial ideal correspondence to the parts in simplicial complexes. Understanding this may help to compute the graph ideals and cut ideals.

Given a graph $G$, how can I compute its graph ideal or cut ideal?

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The Macaulay2 package Graphs does compute several ideals related to graphs. (and you can then process them further with Macaulay2).

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