Tetrahedral curves via graphs and Alexander duality

A tetrahedral curve is a (usually nonreduced) curve in P³ defined by an unmixed, height two ideal generated by monomials. We characterize when these curves are arithmetically Cohen-Macaulay by associating a graph with each curve and, using results from combinatorial commutative algebra and Alexander duality, relating the structure of the complementary graph to the Cohen-Macaulay property.