  For a (finite) partially ordered set (poset) $P$, we call a dominating set
$D$ in the comparability graph of $P$, an order-sensitive dominating set in $P$
if either $x\in D$ or else $a<x<b$ in $P$ for some $a,b\in D$ for every element
$x$ in $P$ which is neither maximal nor minimal, and denote by
$\gamma_{os}(P)$, the least size of an order-sensitive dominating set of $P$.
For every graph $G$ and integer $k\geq 2$, we associate a graded poset
$\mathscr{P}_k(G)$ of height $k$, and prove that
$\gamma_{os}(\mathscr{P}_3(G))=\gamma_{\text{R}}(G)$ and
$\gamma_{os}(\mathscr{P}_4(G))=2\gamma(G)$ hold, where $\gamma(G)$ and
$\gamma_{\text{R}}(G)$ are the domination and Roman domination number of $G$,
respectively. Apart from these, we introduce the notion of a Helly poset, and
prove that when $P$ is a Helly poset, the computation of order-sensitive
domination number of $P$ can be interpreted as a weighted clique partition
number of a graph, the middle graph of $P$. Moreover, we show that the
order-sensitive domination number of a poset $P$ exactly corresponds to the
biclique vertex-partition number of the associated bipartite transformation of
$P$. Finally, we prove that the decision problem of order-sensitive domination
on posets of arbitrary height is NP-complete, which is obtained by using a
reduction from EQUAL-$3$-SAT problem.
