On Hilbert bases of cuts

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A Hilbert basis is a set of vectors X such that the integer cone (semigroup) generated by X is the intersection of the lattice generated by X with the cone generated by X. Define a graph to be (cut) Hilbert if its set of cuts forms a Hilbert basis. We show that the Hilbert property is not closed under edge deletions, subdivisions, nor 2-sums. Furthermore, no graph having K₆-e as a minor is Hilbert. This corrects an error in [M. Laurent. Hilbert bases of cuts. Discrete Math., 150(1-3):257-279 (1996)]. For positive results, we give conditions under which the 2-sum of two graphs produces a Hilbert graph. Using these conditions we show that all H-minor-free graphs are Hilbert, where H is the unique 3-connected graph obtained by uncontracting an edge of K₅. We also establish a relationship between edge deletion and subdivision. Namely, if G' is obtained from a Hilbert graph G by subdividing an edge e two or more times, then G-e is Hilbert if and only if G' is Hilbert.