Resolutions of monomial ideals of projective dimension 1

We show that a monomial ideal I in a polynomial ring S has projective dimension 1 if and only if the minimal free resolution of S/I is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the S/I. We also provide a new characterization of quasi-trees, which we use to give a new proof to a result by Herzog, Hibi, and Zheng which characterizes monomial ideals of projective dimension 1 in terms of quasi-trees.