Almost complete intersection binomial edge ideals and their Rees algebras

Let G be a simple graph on n vertices and JG denote the binomial edge ideal of G in the polynomial ring S=K[x1,…,xn,y1,…,yn]. In this article, we compute the second graded Betti numbers of JG, and we obtain a minimal presentation of it when G is a tree or a unicyclic graph. We classify all graphs whose binomial edge ideals are almost complete intersection, prove that they are generated by a d-sequence and that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees algebra of those binomial edge ideals.