  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 K_6-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 K_5. 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.
