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 $xin D$ or else $a<x<b$ in $P$ for some $a,bin D$ 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 $kgeq 2$, 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 $gamma_{os}(mathscr{P}_4(G))=2gamma(G)$ 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.