Complete intersection toric ideals of oriented graphs and chorded-theta subgraphs

Abstract

Let G = (V, E) be a finite, simple graph. We consider for each oriented graph G O associated to an orientation O of the edges of G, the toric ideal P GO . In this paper we study those graphs with the property that P GO is a binomial complete intersection, for all O. These graphs are called CIO graphs. We prove that these graphs can be constructed recursively as clique-sums of cycles and/or complete graphs. We introduce the chorded-theta subgraphs and their transversal triangles. Also we establish that the CIO graphs are determined by the property that each chorded-theta has a transversal triangle. As a consequence, we obtain that the tournaments hold this property. Finally we explicitly give the minimal forbidden induced subgraphs that characterize these graphs, these families of graphs are: prisms, pyramids, thetas and a particular family of wheels that we call θ−partial wheels.

Matroid Toric Ideals: Complete Intersection, Minors, and Minimal Systems of Generators

“…Therefore, if we take B a basecobase of M and denote by B ′ its complementary base-cobase, then ∆ {B,B ′ } = 252/2 = 126. Let us see now that M has not a minor isomorphic to U 5,10 . Suppose that there exist A, B ⊂ E 1 ∪ E 2 such that U 5,10 ≃ (M\A)/B.…”

“…Since then, they have been extensively studied by several authors. In the context of toric ideals associated to combinatorial structures, the complete intersection property has been widely studied for graphs, see, e.g., [2,22,10]. In this work we address this problem in the context of toric ideals of matroids and prove that there are essentially three matroids whose corresponding toric ideal is a complete intersection; namely, the rank 2 matroids without loops or coloops on a ground set of 4 elements.…”

Acyclic digraphs giving rise to complete intersections

“…The submatrix in rows 1,2 and 4 and columns (1,2), (2,3) and (4,5) has determinant 2. Note also that columns (1,8), (2,9), (2,6) and (1,7) contain a triangular basis with -1s on the diagonal. 7.…”

“…al. [8], [9] have studied complete intersection affine semigroups defined by directed graphs. We will therefore expand the scope of our investigation slightly from Hasse diagrams to directed acyclic graphs.…”

Graphs and complete intersection toric ideals

“…When graphs are not necessarily bipartite there is some recent work by Tatakis and Thoma [28], in the last section we make use of some of their technical results. For directed graphs, the complete intersection property has also been widely studied, see for example [9,11,22].…”