Order-Sensitive Domination in Partially Ordered Sets and Graphs

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 γos(P), the least size of an order-sensitive dominating set of P. For every graph G and integer k⩾2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\geqslant 2$$\end{document}, we associate to G a graded poset Pk(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr{P}}_{k}(G)$$\end{document} of height k, and prove that γos(P3(G))=γR(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{\text {os}}({\mathscr{P}}_{3}(G))=\gamma _{\text {R}}(G)$$\end{document} and γos(P4(G))=2γ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{\text {os}}({\mathscr{P}}_{4}(G))=2\gamma (G)$$\end{document} hold, where γ(G) and γR(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.