Complete intersection toric ideals of oriented graphs and chorded-theta subgraphs
Let G=(V,E) be a finite, simple graph. We consider for each oriented graph \(G_{\mathcal{O}}\) associated to an orientation \({\mathcal{O}}\) of the edges of G, the toric ideal \(P_{G_{\mathcal{O}}}\). In this paper we study those graphs with the property that \(P_{G_{\mathcal{O}}}\) is a binomial complete intersection, for all \({\mathcal{O}}\). These graphs are called \(\text{CI}{\mathcal{O}}\) graphs. We prove that these graphs can be constructed recursively as clique-sums of cycles and/or complete graphs. We introduce chorded-theta subgraphs and some of their properties. Also we establish that the \(\text{CI}{\mathcal{O}}\) graphs are determined by the property that each chorded-theta has a transversal triangle. Finally we explicitly give the minimal forbidden induced subgraphs that characterize these graphs, these families of forbidden graphs are: prisms, pyramids, thetas and a particular family of wheels that we call ?-partial wheels.
complete intersections toric ideals oriented graphs forbidden induced subgraphs