  Fix a positive integer $n$ and consider the bipartite graph whose vertices
are the $3$-element subsets and the $2$-element subsets of
$[n]=\{1,2,\dots,n\}$, and there is an edge between $A$ and $B$ if $A\subset
B$. We prove that the domination number of this graph is
$\binom{n}{2}-\lfloor\frac{(n+1)^2}{8}\rfloor$, we characterize the dominating
sets of minimum size, and we observe that the minimum size dominating set can
be chosen as an independent set. This is an exact version of an asymptotic
result by Balogh, Katona, Linz and Tuza (2021). For the corresponding bipartite
graph between the $(k+1)$-element subsets and the $k$-elements subsets of $[n]$
($k\geq 3$), we provide a new construction for small independent dominating
sets. This improves on a construction by Gerbner, Kezegh, Lemons, Palmer,
P\'alv\"olgyi and Patk\'os (2012), who studied these independent dominating
sets under the name saturating flat antichains.
