Hypergraphs with high projective dimension and 1-dimensional Hypergraphs

We prove a sufficient and a necessary condition for a square-free monomial ideal $J$ associated to a (dual) hypergraph to have projective dimension equal to the minimal number of generators of $J$ minus 2. We also provide an effective explicit procedure to compute the projective dimension of 1-dimensional hypergraphs $\mathcal{H}$ when each connected component contains at most one cycle. An algorithm to compute the projective dimension is also included. Applications of these results are given; they include, for instance, computing the projective dimension of monomial ideals whose associated hypergraph has a spanning Ferrers graph.