Let R=S/I where S=k[T1,…,Tn] and I is a homogeneous ideal in
S. The acyclic closure R⟨Y⟩ of k over R is a DG algebra
resolution obtained by means of Tate's process of adjoining variables to kill
cycles. In a similar way one can obtain the minimal model S[X], a DG algebra
resolution of R over S. By a theorem of Avramov there is a tight connection
between these two resolutions. In this paper we study these two resolutions
when I is the edge ideal of a path or a cycle. We determine the behavior of
the deviations εi(R), which are the number of variables in
R⟨Y⟩ in homological degree i. We apply our results to the
study of the k-algebra structure of the Koszul homology of R