Separating hyperplanes of edge polytopes

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