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 θ-partial wheels.