On circuit binomials of toric ideals of weighted oriented graphs

In this work, we classify the circuit binomials of any weighted oriented graph $D$ and we explicitly compute the circuit binomials of $D$ in terms of the minors of the incidence matrix of $D$. We show that the circuit binomials of any weighted oriented graph $D$ are the primitive binomials corresponding to one of the classes: (i) a balanced cycle, (ii) two unbalanced cycles sharing a vertex, (iii) two unbalanced cycles connected by a path, (iv) two unbalanced cycles sharing a path. We explicitly prove a formula for the primitive binomial generator of the toric ideal $I_D$ in terms of the minors of the incidence matrix of $D$, where $D$ is as in (i), (ii), (iii) and (iv). Thus we explicitly compute all the circuit binomials ${\text{C}}_D$ of any weighted oriented graph $D$. If $D$ is a weighted oriented graph which has at most two unbalanced cycles such that no two balanced cycles share a path in $D$ and no balanced cycle in $D$ shares an edge with the path which connects the two unbalanced cycles in $D$ if it exists, then we show that $I_D$ is a strongly robust circuit ideal and it has complete intersection initial ideal. For this class of ideals, we explicitly compute the Betti numbers.