Quadruple Roman domination in graphs

Amjadi, Jafar; Khalili, N.

doi:10.1142/s1793830921501305

Cited by 9 publications

(3 citation statements)

References 18 publications

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“…The triple Roman domination, by Ahangar et al [3] and the quadruple Roman domination in graphs, by Amjadi et. al [4,5].…”

Section: Introductionmentioning

confidence: 99%

The [k]-multiple Roman domination in Graphs

Valenzuela-Tripodoro,

Mateos-Camacho,

Cera

et al. 2023

Preprint

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In 2016, Beeler et al. introduced the double Roman domination as a variation of Roman domination and recently Abdollahzadeh et al. (resp. Amjadi et al.) initiated the study of triple (resp. quadruple) Roman domination in graphs. Given any labeling of the vertices of a graph, AN(v) stands for the set of neighbors of a vertex v having a positive label. In this paper we generalize these concepts by studying the [k]-multiple Roman domination functions ([k]-MRDF) in graphs which coincides with the previous versions when 2 ≤ k ≤ 4. Namely, f is a [k]-MRDF if f(N[v]) ≥ k + |AN(v)| for all v. We approach the complexity of the associate decision problem and we also give sharp bounds and exact values for several classes of graphs.

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“…The triple Roman domination, by Ahangar et al [3] and the quadruple Roman domination in graphs, by Amjadi et. al [4,5].…”

Section: Introductionmentioning

confidence: 99%

The [k]-multiple Roman domination in Graphs

Valenzuela-Tripodoro,

Mateos-Camacho,

Cera

et al. 2023

Preprint

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“…The [k]-Roman domination number was introduced by Abdollahzadeh Ahangar et al in [1]. The case k = 1 is Roman domination which was introduced by Cockyne et al in [13], the case k = 2 is the double Roman domination which was investigated in [9], the case k = 3 is triple Roman domination number (TRD-number) and has been studied in [1,2,16], and the case k = 4 is the Quadruple Roman domination and has been investigated in [4]. The literature on Roman domination and its variants has been detailed in two chapters of the books and a surveys paper (see [10][11][12]).…”

Section: Introductionmentioning

confidence: 99%

Triple Roman domination subdivision number in graphs

Amjadi¹,

Sadeghi²

2022

CSJM

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For a graph $G=(V, E)$, a triple Roman domination function is a function $f: V(G)\longrightarrow\{0, 1, 2, 3, 4\}$ having the property that for any vertex $v\in V(G)$, if $f(v)<3$, then $f(\mbox{AN}[v])\geq|\mbox{AN}(v)|+3$, where $\mbox{AN}(v)=\{w\in N(v)\mid f(w)\geq1\}$ and $\mbox{AN}[v]=\mbox{AN}(v)\cup\{v\}$. The weight of a triple Roman dominating function $f$ is the value $\omega(f)=\sum_{v\in V(G)}f(v)$. The triple Roman domination number of $G$, denoted by $\gamma_{[3R]}(G)$, equals the minimum weight of a triple Roman dominating function on $G$. {\em The triple Roman domination subdivision number} $\mbox{sd}_{\gamma_{[3R]}}(G)$ of a graph $G$ 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 triple Roman domination number. In this paper, we first show that the decision problem associated with $\mbox{sd}_{\gamma_{[3R]}}(G)$ is NP-hard and then establish upper bounds on the triple Roman domination subdivision number for arbitrary graphs.

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“…The case k = 1 is the usual Roman domination, the case k = 2 is called double Roman domination and were studied in several papers [11][12][13][14][15][16][17][18][19][20] while the case k = 3 is called the triple Roman domination and were studied in several papers [10,21]. The case k = 4 is called quadruple Roman domination and were studied in [22].…”

Section: Introductionmentioning

confidence: 99%

Quadruple Roman Domination in Trees

et al. 2021

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This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,…,k+1} if for any vertex u∈V with f(u)<k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.

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Quadruple Roman domination in graphs

Cited by 9 publications

References 18 publications

The [k]-multiple Roman domination in Graphs

The [k]-multiple Roman domination in Graphs

Triple Roman domination subdivision number in graphs

Quadruple Roman Domination in Trees

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