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Let G be a graph on [n] and J_G be the binomial edge ideal of G in the polynomial ring S=K[x_1,...,x_n,y_1,...,y_n]. In this paper we investigate some topological properties of a poset associated to the minimal primary decomposition of J_G. We show that this poset admits some specific subposets which are contractible. This in turn, provides some interesting algebraic consequences. In particular, we characterize all graphs G for which depth S/J_G=4.
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