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A minimal sufficient adjustment set is a sufficient adjustment set none of whose proper subsets is sufficient. Minimality is by set inclusion and does not imply minimum cardinality or uniqueness.

Among observed pretreatment variables, every sufficient set must contain $M_1$ to blockand $Z_3$ to blockConditioning on $Z_3$ opens the collider onso $M_3$ must also be included. The resulting minimal sufficient adjustment set isIt blocks every path from $Z_1$ to $Y$ that remains after removing $A\to Y$, while the open pathpreserves instrument relevance. Adding $M_2$ blocks no required relevance path, sois also sufficient.

There are no others. In particular, adding $Z_2$ blocks the displayed relevance path. Without $M_2$, conditioning on $Z_2$ also openswhich violates independence; adding $M_2$ closes that path but leaves no open path from $Z_1$ to $A$. Hence the complete list is $X_0$ and $X_1$.

Let $M$ be a Markov blanket of the treatment $A$ inside $(A,X)$, and write the remaining adjustment variables as $N=X\setminus M$. By definition,The sufficiency of $X=(M,N)$ givesApply the contraction axiom for conditional independence with first variable $A$, second variable $N$, third variable $Y(a)$, and conditioning variable $M$. It givesThe decomposition axiom for conditional independence then yields $A\perp Y(a)\mid M$. Hence every Markov blanket of $A$ in $(A,X)$ is itself a sufficient adjustment set.