= Solution
To include all momentum-dependent contributions to the <two-point function>, put one external line at each <interaction vertex>. Let $r$ be the number of lines connecting the vertices, and $t_A,t_B$ their numbers of <tadpole diagrams>. For vertex valences $v_A,v_B\in\{4,6\}$,
$$
r+2t_A=v_A-1,\qquad r+2t_B=v_B-1.
$$
The integer $r$ is odd, and $r=1$ gives a <one-particle-reducible Feynman diagram> and is excluded. Thus the complete list is
* $v_A=v_B=4$: $r=3$, $t_A=t_B=0$, the <sunset diagram>.
* $v_A=4,v_B=6$: $r=3$, $t_A=0,t_B=1$, the <sunset diagram> with one <tadpole diagram>.
* $v_A=v_B=6$: $r=3$, $t_A=t_B=1$, the <sunset diagram> with one <tadpole diagram> at each vertex.
* $v_A=v_B=6$: $r=5$, $t_A=t_B=0$, the <five-line banana integral> topology.
The coefficient of the mixed graph includes the reflected external attachment as well as its reciprocal <Feynman-diagram symmetry factor>. If both external lines attach to the same <interaction vertex>, no external <momentum> enters the connecting internal lines, and that <Feynman diagram> cannot contribute to <wave-function renormalization>.
For incoming external <momentum> $p$, assign the connecting lines <momenta> $k_1,\ldots,k_{r-1},p-\sum_jk_j$. Define
$$
B_{r,M}(p)=\int\prod_{j=1}^{r-1}\frac{d^3k_j}{(2\pi)^3}
\left[\prod_{j=1}^{r-1}D(k_j)\right]D\!\left(p-\sum_{j=1}^{r-1}k_j\right),\qquad r=3,5.
$$
The <momentum space> <Feynman rules> give, in the order listed,
$$
\boxed{\begin{aligned}
&C_{44}(-ig_1)^2B_{3,M}(p),\\
&C_{46}(-ig_1)(-ig_2)T_MB_{3,M}(p),\\
&C_{66,3}(-ig_2)^2T_M^2B_{3,M}(p),\\
&C_{66,5}(-ig_2)^2B_{5,M}(p).
\end{aligned}}
$$
Each $C$ is a reciprocal <Feynman-diagram symmetry factor>, including any equivalent external attachments. There are respectively $2,3,4,4$ independent <loop momenta>.
\b[If “contribute to wavefunction renormalization” means requiring a divergent kinetic counterterm after subtraction of all subdivergences, only the last topology qualifies.] The first three still have nontrivial external <momentum> dependence and give finite contributions to a normalization condition; their raw divergent factors arise from <mass counterterms> and <interaction vertex> <counterterms>. Keeping all four here makes this distinction explicit. The <ultraviolet divergence> analysis in <3/d> establishes it directly.
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