I'm not sure exactly what the question is, but let me comment that lots of Feynman graphs with lots of different rules for labelling the vertices come up in QFT, as explained by Theo.  From the QFT point of view, the labelling you're describing is not particularly common; you have infinitely many terms in your Lagrangian, for instance.  So I would say that the answer is no, stable graphs do not come up much beyond the cases that are clearly related to Gromov-Witten theory or other string theories.