On the extremal Betti numbers of the binomial edge ideal of closed graphs

We study the equality of the extremal Betti numbers of the binomial edge ideal $J_G$ and those of its initial ideal ${rm in}(J_G)$ of a closed graph $G$. We prove that in some cases there is an unique extremal Betti number for ${rm in}(J_G)$ and as a consequence there is an unique extremal Betti number for $J_G$ and these extremal Betti numbers are equal