Morse resolutions of powers of square-free monomial ideals of projective dimension one

Let $I$ be a square-free monomial ideal $I$ of projective dimension one. Starting with the Taylor complex on the generators of $I^r$, we use Discrete Morse theory to describe a CW complex that supports a minimal free resolution of $I^r$. To do so, we concretely describe the acyclic matching on the faces of the Taylor complex.