  Let $G=(V,E)$ be a finite, simple graph. We consider for each oriented graph
$G_{\cal O}$ associated to an orientation ${\cal O}$ of the edges of $G$, the
toric ideal $P_{G_{\cal O}}$. In this paper we study those graphs with the
property that $P_{G_{\cal O}}$ is a binomial complete intersection, for all
${\cal O}$. These graphs are called $\text{CI}{\cal O}$ 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 $\text{CI}{\cal O}$ 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 $\theta-$partial wheels.
