One and two leg diagrams are not interactions, as pointed out by Felicia in the comments.
A one leg diagram corresponds to a "tadpole", which have to vanish for stability reasons. Physically, these diagrams would correspond to processes where particles just pop out of the vacuum, and therefore if they didn't vanish they would imply that the vacuum is unstable. Mathematically, these diagrams vanish when the background fields satisfy the classical equations of motion. For instance, in the standard model, tadpole terms involving the Higgs vanish when the Higgs VEV is at a minimum of the potential.
A two leg diagram is a propagator, which is not an interaction -- one particle goes on, one goes out.
Interactions are described by vertices with three or more particles.