To amplify Theo's comment slightly: The graphs that show up in Feynman diagram perturbation theory are stable because the physicists use a different accounting system for the genus zero vertices with 0,1, or 2 edges. The diagrams with these graphs don't show up in the perturbation series because the physical effects they represent are the situation one is perturbing away from. A genus zero graph with two edges is an order $\mathcal{O}(\hbar^0)$ correction to the propagator. Likewise, one edge gives a tadpole correction to the expectation value of the field (usually gotten rid of by redefining the field), and zero edges a correction to the vacuum energy (usually set to zero by convention).

To amplify Theo's comment slightly: The graphs that show up in Feynman diagram perturbation theory are stable because the physicists use a different accounting system for the genus zero vertices with 0,1, or 2 edges. The diagrams with these graphs don't show up in the perturbation series because the physical effects they represent are the situation one is perturbing away from. A genus zero graph with two edges is an order $\mathcal{O}(\hbar^0)$ correction to the propagator. Likewise, one edge gives a tadpole correction to the expectation value of the field (usually gotten rid of by redefining the field), and zero edges a correction to the vacuum energy (usually set to zero by convention).