  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.
