Separating hyperplanes of edge polytopes

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.