On robust toric ideals of weighted oriented graphs

Abstract: In this work, we study the equivalence of robustness, strongly robustness, generalized robustness, and weakly robustness properties of toric ideals of weighted oriented graphs. For any weighted oriented graph $D$, if its toric ideal $I_D$ is generalized robust or weakly robust, then we show that $D$ has no subgraphs of certain structures. We prove the equality of Graver basis, Universal Gr\"obner basis, reduced Gr\"obner basis with respect to degree lexicographic order of the toric ideals $I_D$ of weighted oriented graphs $D=\C_1\cup_P \cdots \cup_P \C_n$ consist of cycles $\C_1\ldots,\C_n$ that share a path $P$. As a consequence, we show that (i) $I_{D}$ is robust iff $I_{D}$ is strongly robust; (ii) $I_{D}$ is generalized robust iff $I_{D}$ is weakly robust. If at most two of the cycles $\C_i$ in $D$ are unbalanced, then the following statements are equivalent: (i) $I_{D}$ is strongly robust; (ii) $I_{D}$ is robust; (iii) $I_{D}$ is generalized robust; (iv) $I_{D}$ is weakly robust; (v) $D$ has no subgraphs of types $D_{1}$ and $D_{2}$, where $D_1$ is a weighted oriented graph consisting of two balanced cycles share an edge in $D$, and $D_2$ is a weighted oriented graph consisting of three cycles that share an edge\ such that one cycle is balanced and the rest two are unbalanced cycles, as in figure \ref{fig2}. We explicitly determine the Graver basis of the toric ideal of two balanced cycles sharing a path.