Strongly robustness of toric ideals of weighted oriented cycles sharing a vertex
Ramakrishna Nanduri, Tapas Kumar Roy
TL;DR
This work analyzes strong robustness of toric ideals associated with vertex-weighted oriented graphs built from cycles sharing a common vertex, formalized via toric ideals $I_D$ and $I_{D'}$. For graphs $D$ and $D'$ as described, the authors prove that $I_D$ and $I_{D'}$ are strongly robust, hence robust, with the Graver basis coinciding with the minimal generating set of binomials. In the special case where $D$ consists of three unbalanced cycles sharing a vertex, they derive an explicit unique minimal generating set of primitive binomials, described in terms of minors of the incidence matrix. The results extend to broader classes by showing robustness is preserved under attaching disjoint cycles via paths and by clarifying when circuit- and weight-independence phenomena occur, contributing to the broader understanding of toric ideals of weighted oriented graphs.
Abstract
In this article, we study the strongly robust property of toric ideals of weighted oriented graphs. Let $D$ be a weighted oriented graph consists of weighted oriented cycles (balanced or unbalanced) sharing a single vertex $v$ and $D^{\prime}$ be a weighted oriented graph consists of $D$ and a finite number of disjoint cycles such that each of these cycles is connected by a path at the sharing vertex $v$ of $D$. Then we show that the toric ideals $I_D,I_{D^{\prime}}$ of $D$ and $D^{\prime}$ respectively, are strongly robust and hence robust. That is, for the toric ideal $I_D$, of $D$, its Graver basis is a minimal generating set of $I_D$. If $D$ is a weighted oriented three cycles sharing a single vertex, then We explicitly give a unique minimal generating set of primitive binomials of $I_D$ in terms of minors of the incidence matrix of $D$.