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<p>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≥ 2, we associate to G a graded poset P_k(G) of height k, and prove that γ_os(P_3(G))=γ_R(G) and γ_os(P_4(G))=2γ(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.<br></p> |
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