  Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field
$\mathbb{K}$, one can associate it with its LCM-lattice and its hypergraph. In
this short note, we establish the connection between the LCM-lattice and the
hypergraph, and in doing so we provide a sufficient condition for removing
higher dimension edges of the hypergraph without impacting the projective
dimension of the square-free monomial ideal. We also offer algorithms to
compute the projective dimension of a class of square-free monomial ideals
built using the new result and previous results of Lin-Mantero.
