Toric Geometry of Series-Parallel Graphs

Abstract

Let G be a graph and $\mathbb{K}$ be a field. We associate to G a projective toric variety $X_G$ over $\mathbb{K}$, the cut variety of the graph G. The cut ideal $I_G$ of the graph G is the ideal defining the cut variety. We show that, if G is a subgraph of a subdivision of a book or an outerplanar graph, then the minimal generators are quadrics. Furthermore, we describe the generators of the cut ideal of a subdivision of a book.