Forgive me if this problem is known by another name, I do not know any references for it.
Symmetric deduction. An equation $e \in E$ is a subset of variables $V$ such that knowing $|e| - 1$ of the variables allows you to deduce the missing one.
Minimal cover under symmetric deduction. Does there exist a set of $k$ variables such that their closure under symmetric deduction with equations $E$ equals all variables $V$?
This problem is proven NP-complete in this question. There is a limited version of this problem:
Minimal cover under symmetric $d$-deduction. Same as before, except $\forall e\in E: |e| \leq d$. That is, each equation has degree $d$ or lower.
This problem is trivial for $d = 2$ (choose one variable per connected component), and NP-hard for $d = 4$ as shown by Thinh D. Nugyen here.
What about $d = 3$?