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 GO associated to an orientation O of the edges of G, the toric ideal PGO. In this paper we study those graphs with the property that PO 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 chorded-theta subgraphs and some of their properties. Also we establish that the CIO 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 θ-partial wheels. © 2013 Springer Science+Business Media New York.