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We prove a sufficient and a necessary condition for a square-free monomialideal $J$ associated to a (dual) hypergraph to have projective dimension equalto the minimal number of generators of $J$ minus 2. We also provide aneffective explicit procedure to compute the projective dimension of1-dimensional hypergraphs $\mathcal{H}$ when each connected component containsat most one cycle. An algorithm to compute the projective dimension is alsoincluded. Applications of these results are given; they include, for instance,computing the projective dimension of monomial ideals whose associatedhypergraph has a spanning Ferrers graph.

Hypergraphs with high projective dimension and 1-dimensional Hypergraphs