In this paper we study the universal stability of undirected graphs in the adversarial queueing model for packet routing. In this setting, packets must be injected in some edge and have to traverse a path before leaving the system. Restrictions on the allowed types of path that packets must traverse provide different packet models. We consider three natural models, and provide polynomial time algorithms for testing universal stability on them. In the three cases, we obtain a different characterization, in terms of forbidden subgraphs, thus showing that slight variations lead to non-equivalent models.We extend those results to show that universal stability of digraphs, in the case in which packets follow directed paths without repeating vertices, can be decided in polynomial time.All the instability results are obtained for the \NTGLIS protocol. Therefore, the property of universal stability is equivalent to \NTGLIS-stability, in all the cases.