Separating hyperplanes of edge polytopes

TL;DR

This paper investigates the conditions under which edge polytopes of graphs can be decomposed via hyperplanes, providing algorithms, normality criteria, and analysis of toric ideals related to such decompositions.

Contribution

It introduces an algorithm for deciding decomposability of edge polytopes and studies the preservation of normality and quadratic generation of toric ideals under decomposition.

Findings

An algorithm for decomposability decision

Normality preserved under decomposition

Toric ideal quadratic generation behavior analyzed

Abstract

Let $G$ be a finite connected simple graph with $d$ vertices and let $\Pc_G \subset \RR^d$ be the edge polytope of $G$. We call $\Pc_G$ \emph{decomposable} if $\Pc_G$ decomposes into integral polytopes $\Pc_{G^+}$ and $\Pc_{G^-}$ via a hyperplane. In this paper, we explore various aspects of decomposition of $\Pc_G$: we give an algorithm deciding the decomposability of $\Pc_G$, we prove that $\Pc_G$ is normal if and only if both $\Pc_{G^+}$ and $\Pc_{G^-}$ are normal, and we also study how a condition on the toric ideal of $\Pc_G$ (namely, the ideal being generated by quadratic binomials) behaves under decomposition.