In information theory, structural system constraints are frequently described in the form of a directed acyclic graphical model (DAG). This paper addresses the question of classifying DAGs up to an isomorphism. By considering Gaussian densities, the question reduces to verifying equality of certain algebraic varieties. A question of computing equations for these varieties has been previously raised in the literature. Here it is shown that the most natural method adds spurious components with singular principal minors, proving a conjecture of Sullivant. This characterization is used to establish an algebraic criterion for isomorphism, and to provide a randomized algorithm for checking that criterion. Results are applied to produce a list of the isomorphism classes of tree models on 4 and 5 nodes.National Science Foundation (U.S.) (CAREER Award Grant Agreement CCF-12-53205)Siemens Corporation (MIT Energy Initiative)