关于割的Hilbert基
On Hilbert bases of cuts
摘要：
希尔伯特基是一组向量X，使得由X生成的整数锥（半群）是由X生成的格与由X生成的锥的交集。定义一个图是（割）希尔伯特，如果它的割集形成一个希尔伯特基。我们证明了Hilbert性质在边缘删除、细分和2-和下都不是封闭的。而且，没有一个图的子图是Hilbert的.这纠正了[M]中的错误。劳伦特希尔伯特截基。离散数学，150（1-3）：257-279（1996）]。对于肯定的结果，我们给出了两个图的2-和产生Hilbert图的条件。利用这些条件，我们证明了所有无H-子图的图都是Hilbert图，其中H是通过对K_5的一条边进行非收缩而得到的唯一3-连通图.我们还建立了边删除和细分之间的关系。也就是说，如果G'是从一个Hilbert图G通过将一条边e细分两次或更多次而得到的，则G-e是Hilbert当且仅当G'是Hilbert。
英文摘要：
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.
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