Oriented complex-scalar quartic graph factors

For interaction $-\lambda {\varphi }^2({\varphi }^{\ast }{)}^2/4$, the vertex factor is $-i\lambda$. A two-line bubble carrying both internal arrows in the same direction has symmetry factor two. Oppositely directed lines cannot be interchanged as an oriented graph, so the two mixed external channels have factor one. A directed tadpole inserted on a labelled external leg also has factor one. Thus the connected four-point function at second order includes three bubble attachments and four external-leg insertions; retaining only one-particle-irreducible bubbles would compute a different object. These factors do not depend on whether the mixed free kernel is $D_F$ or a consistently rescaled multiple of it.