On Hilbert bases of cuts

TL;DR

This paper investigates the properties of Hilbert bases in graph cuts, identifying conditions under which graphs are Hilbert and correcting previous misconceptions about their closure properties.

Contribution

It corrects an error in prior work, establishes new conditions for Hilbert graphs, and characterizes Hilbert properties in minor-free graphs and under edge subdivision.

Findings

Hilbert property is not closed under edge deletions, subdivisions, or 2-sums.

No graph with K_6-e as a minor is Hilbert.

All H-minor-free graphs are Hilbert, where H is a specific 3-connected graph.

Abstract

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.