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On the Roman domination subdivision number of a graph

Author

Listed:
  • J. Amjadi

    (Azarbaijan Shahid Madani University)

  • R. Khoeilar

    (Azarbaijan Shahid Madani University)

  • M. Chellali

    (University of Blida)

  • Z. Shao

    (Guangzhou University)

Abstract

A Roman dominating function (RDF) of a graph G is a labeling $$f:V(G)\longrightarrow \{0,1,2\}$$ f : V ( G ) ⟶ { 0 , 1 , 2 } such that every vertex with label 0 has a neighbor with label 2. The weight of an RDF is the sum of its functions values over all vertices, and the Roman domination number of G is the minimum weight of an RDF of G. The Roman domination subdivision number $$\mathrm {sd}_{\gamma _{R}}(G)$$ sd γ R ( G ) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the Roman domination number of G. In this paper, we present a new upper bound on the Roman domination subdivision number by showing that for every connected graph G of order at least three, $$\begin{aligned} \mathrm {sd}_{\gamma _{R}}(G)\le 3+\min \{\deg _2(v)\mid v\in V\;\mathrm {and} \;d(v)\ge 2\}, \end{aligned}$$ sd γ R ( G ) ≤ 3 + min { deg 2 ( v ) ∣ v ∈ V and d ( v ) ≥ 2 } , where $$\deg _2(v)$$ deg 2 ( v ) is the number of vertices of G at distance 2 from vertex v. Moreover, we show that the decision problem associated with $$\mathrm {sd}_{\gamma _{R}}(G)$$ sd γ R ( G ) is NP-hard for bipartite graphs.

Suggested Citation

  • J. Amjadi & R. Khoeilar & M. Chellali & Z. Shao, 2020. "On the Roman domination subdivision number of a graph," Journal of Combinatorial Optimization, Springer, vol. 40(2), pages 501-511, August.
  • Handle: RePEc:spr:jcomop:v:40:y:2020:i:2:d:10.1007_s10878-020-00597-x
    DOI: 10.1007/s10878-020-00597-x
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    Cited by:

    1. Guoliang Hao & Seyed Mahmoud Sheikholeslami & Mustapha Chellali & Rana Khoeilar & Hossein Karami, 2021. "On the Paired-Domination Subdivision Number of a Graph," Mathematics, MDPI, vol. 9(4), pages 1-9, February.
    2. Enqiang Zhu, 2021. "An Improved Nordhaus–Gaddum-Type Theorem for 2-Rainbow Independent Domination Number," Mathematics, MDPI, vol. 9(4), pages 1-10, February.

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