Robust Graph Ideals

Abstract

Let I be a toric ideal. We say I is robust if its universal Gröbner basis is a minimal generating set. We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely characterize all graphs which give rise to robust ideals. Our characterization shows that robustness can be determined solely in terms of graphtheoretic conditions on the set of circuits of G.Theorem 1.2. I G is robust if and only if the following conditions are satisfied.R1: No circuit of G has an even chord, R2: No circuit of G has a bridge, R3: No circuit of G contains an effective crossing, and R4: No circuit of G shares exactly one edge (and no other vertices) with another circuit such that the shared edge is part of a cyclic block in both circuits.

Bouquet algebra of toric ideals

“…It is also well known that Lawrence liftings are not the only matrices with this property. For example, it was shown to hold for 2-regular uniform hypergraphs by Gross and Petrović in [23], and for robust toric ideals of graphs by Boocher et al in [10]. Furthermore, such examples can have both mixed and non-mixed bouquets, as in Example 4.3 (b).…”

“…It follows from Theorem 3.7 that I A is a robust toric ideal if and only if I A B is a robust toric ideal. In particular, using the same strategy described above for generic lattice ideals, we can construct robust toric ideals that are different from the ones considered in [9,10] which, in fact, correspond to toric ideals of graphs and toric ideals generated by degree two binomials. Using again Theorem 3.7 we also have that I A is generalized robust toric ideal if and only if I A B is a generalized robust toric ideal (see [37] for the definition of generalized robust toric ideal).…”

“…The more technical name for strongly robust ideals, which are introduced in Section 4, is ∅-Lawrence. However, it is conjectured that all robust toric ideals are minimally generated by their Graver bases, see [10,Question 6.1], in other words, that all robust toric ideals are strongly robust. As we will see below, bouquets are exactly what can be used to classify toric ideals that belong to this class.…”

Generalized robust toric ideals

“…In the first part, we study the generalized robustness on toric ideals of a graph G. The results of this part are inspired and guided by the work of [3] in order to give theorems that fully characterize the generalized robust toric ideals of graphs. The papers [20], [22] and [24] describe the Markov basis, the Graver basis, the universal Gröbner basis and the set of the circuits for a toric ideal arising from a graph.…”

“…In section 2, we analyze all these notions more explicitly. Applying this knowledge on the work of Boocher et al (see [3]), we are allowed to provide the study of the generalized robustness of graphs, with theorems of the same structure as theirs. In section 3, we first prove that a toric graph ideal is generalized robust if and only if its universal Markov basis is equal to the Graver basis of the ideal, see Theorem 3.4.…”