(Dual) hypergraphs have been used by Kimura, Rinaldo and Terai to characterize squarefree monomial ideals J with pd(R/J) ≤ μ(J) - 1, i.e. whose projective dimension equals the minimal number of generators of J minus 1. In this paper, we prove sufficient and necessary combinatorial conditions for pd(R/J) ≤ μ(J) - 2. The second main result is an effective explicit procedure to compute the projective dimension of a large class of 1-dimensional hypergraphs (the ones in which every connected component contains at most one cycle). An algorithm to compute the projective dimension is also provided. 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.
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