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On a Relation between the Perfect Roman Domination and Perfect Domination Numbers of a Tree

Author

Listed:
  • Zehui Shao

    (Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China)

  • Saeed Kosari

    (Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China)

  • Mustapha Chellali

    (LAMDA-RO Laboratory, Department of Mathematics, University of Blida, Blida B.P. 270, Algeria)

  • Seyed Mahmoud Sheikholeslami

    (Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran)

  • Marzieh Soroudi

    (Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran)

Abstract

A dominating set in a graph G is a set of vertices S ⊆ V ( G ) such that any vertex of V − S is adjacent to at least one vertex of S . A dominating set S of G is said to be a perfect dominating set if each vertex in V − S is adjacent to exactly one vertex in S . The minimum cardinality of a perfect dominating set is the perfect domination number γ p ( G ) . A function f : V ( G ) → { 0 , 1 , 2 } is a perfect Roman dominating function (PRDF) on G if every vertex u ∈ V for which f ( u ) = 0 is adjacent to exactly one vertex v for which f ( v ) = 2 . The weight of a PRDF is the sum of its function values over all vertices, and the minimum weight of a PRDF of G is the perfect Roman domination number γ R p ( G ) . In this paper, we prove that for any nontrivial tree T , γ R p ( T ) ≥ γ p ( T ) + 1 and we characterize all trees attaining this bound.

Suggested Citation

  • Zehui Shao & Saeed Kosari & Mustapha Chellali & Seyed Mahmoud Sheikholeslami & Marzieh Soroudi, 2020. "On a Relation between the Perfect Roman Domination and Perfect Domination Numbers of a Tree," Mathematics, MDPI, vol. 8(6), pages 1-13, June.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:6:p:966-:d:370709
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    References listed on IDEAS

    as
    1. Abdollah Alhevaz & Mahsa Darkooti & Hadi Rahbani & Yilun Shang, 2019. "Strong Equality of Perfect Roman and Weak Roman Domination in Trees," Mathematics, MDPI, vol. 7(10), pages 1-13, October.
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