Order-sensitive domination in partially ordered sets

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.