If $[n]:=\{1,2,\ldots, n\}$ for some $n\in\mathbb{N}$, then the *hypercube digraph* of dimension $n$, denoted $H\_n$, is the graph whose set of vertices is the power-set $\wp([n])$ where two vertices $U,V\subseteq [n]$ are adjacent and oriented as $\langle U, V\rangle$ if and only if there exists some $q\in [n]$ such that $U=V\setminus \{q\}$.


A *cut* $C$ of $H\_n$ is a subset of the vertex set $C\subseteq \wp([n])\setminus \{\emptyset, [n]\}$ such that that (1) it does not contains neither the bottom vertex $\emptyset$ nor the top vertex $[n]$ (2) there exists no directed path from $\emptyset$ to $[n]$ in the subgraph of $H\_n$ induced by $\wp([n])\setminus C$, i.e. there is no directed path in $H\_n$ that goes from the bottom vertex $\emptyset$ to the top vertex $[n]$ simultaneously avoiding all vertices of $C$. 


A cut $C$ is minimal when for every $S\subsetneq C$ we have that $S$ is not a cut.


**Q.** Is it known how to count and/or generate the family of minimal cuts on the directed hypercube graphs ?


**Q.** Let $\gamma(n)$ be the number of minimal cuts of $H\_n$, is this integer sequence already known in the literature ? 


**Q.** Is it anything known on minimal cuts ? 



Directed Hypercube Minimal Cuts
I do not know the answer to your first two questions, but here is what I was able to find relevant to your third question.



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What you call the hypercube digraph is the Hasse diagram of $B\_n$ the Boolean algebra (or Boolean lattice). Using this keyword may help you find some literature on the subject.


In [this paper](http://www.math.uiuc.edu/~z-furedi/PUBS/furedi_griggs_kleitman_cutsets.pdf) by Furedi, Griggs, and Kleitman a minimal cutset of the Boolean algebra with almost all the members is found. That is they show existence of a minimal cut set in $B\_n$ of size $c(n)$ where $\lim\_{n \to \infty} c(n)/2^n = 1$.


That paper is from 1988, and I was not able to find anything more recent. 


It seems difficult to enumerate all the minimal cutsets considering there is such a wide range of sizes. There are the obvious cutsets of size $\binom{n}{k}$ taking all the $k$ element subsets. Thus there is one of size $n$ taking all the $1$ element subsets. Also, the result in the paper above gives an "almost" explicit description of the cutset of size $c(n)$. So, what theis large cutset looks like is not exactly known (at least at the time of this paper).


Though maybe more is known today. I was not able to find anything on OEIS. It would be interesting to see some data at least for small $n$.


