On robust toric ideals of weighted oriented graphs

Abstract

In this work, we study the equivalence of various 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 $D$ does not have forbidden subgraphs $D_1,D_2$ of certain structures. We give a significant class of weighted oriented graphs $D$ whose toric ideals $I_D$ have the following equivalence. (i) $I_{D}$ is strongly robust (equivalently, $I_{D}$ is robust); (ii) $I_{D}$ is generalized robust (equivalently, $I_{D}$ is weakly robust); (iii) $D$ does not have subgraphs equal to $D_{1}$ and $D_{2}$.