Order-sensitive domination in partially ordered sets and graphs

Civan Y., Deniz Z., Yetim M. A.

ORDER, vol.40, no.1, pp.157-172, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 40 Issue: 1
  • Publication Date: 2023
  • Doi Number: 10.1007/s11083-022-09599-2
  • Journal Name: ORDER
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.157-172
  • Keywords: Domination, Partially ordered set, Order-sensitive, Comparability, Roman domination, Biclique Vertex-partition
  • Süleyman Demirel University Affiliated: Yes

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≥ 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.