Almost complete intersection binomial edge ideals and their Rees algebras.

Let G be a simple graph on n vertices and J G denote the binomial edge ideal of G in the polynomial ring S = K [ x 1 , ... , x n , y 1 , ... , y n ]. In this article, we compute the second graded Betti numbers of J G , 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. [ABSTRACT FROM AUTHOR]