Hilbert bases of cuts Original Research Article

Let X be a set of vectors in Rm. X is said to be a Hilbert base if every vector in Rm which can be written both as a linear combination of members of X with nonnegative coefficients and as a linear combination with integer coefficients can also be written as a linear combination with nonnegative integer coefficients. Denote by H the collection of the graphs whose family of cuts is a Hilbert base. It is known that K5 and graphs with no K5-minor belong to H and that K6 does not belong to H. We show that every proper subgraph of K6 belongs to H and that every graph from H does not have K6 as a minor. We also study how the class H behaves under several operations.