Order-Sensitive Domination in Partially Ordered Sets and Graphs

Abstract

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 ∈ D or else a < x < b in P for some a,b ∈ D for every element x in P which is neither maximal nor minimal, and denote by γ_os(P), the least size of an order-sensitive dominating set of P. For every graph G and integer $k\geqslant 2$, we associate to G a graded poset ${\mathscr{P}}_{k}(G)$ of height k, and prove that $\gamma _{\text {os}}({\mathscr{P}}_{3}(G))=\gamma _{\text {R}}(G)$ and $\gamma _{\text {os}}({\mathscr{P}}_{4}(G))=2\gamma (G)$ hold, where γ(G) and γ_R(G) are the domination and Roman domination number of G respectively. 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.