The Perfect Roman Domination Number of the Cartesian Product of Some Graphs

Almulhim, Ahlam; Akwu, Abolape Deborah; Subaiei, Bana Al

doi:10.1155/2022/1957027

Cited by 3 publications

(1 citation statement)

References 17 publications

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“…Since then, over 100 papers on Roman domination and its variants have been published. Perfect Roman domination [5,6], Italian domination [7,8], perfect Italian domination [9], double-Roman domination [10], perfect double-Roman domination [11], double-Italian domination [12], total Roman domination [13,14], vertex-edge Roman domination [15], and vertex-edge perfect Roman domination [16] are a few examples of Roman domination variants.…”

Section: Introductionmentioning

confidence: 99%

Total Perfect Roman Domination

Almulhim

2023

Symmetry

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A total perfect Roman dominating function (TPRDF) on a graph G=(V,E) is a function f from V to {0,1,2} satisfying (i) every vertex v with f(v)=0 is a neighbor of exactly one vertex u with f(u)=2; in addition, (ii) the subgraph of G that is induced by the vertices with nonzero weight has no isolated vertex. The weight of a TPRDF f is ∑v∈Vf(v). The total perfect Roman domination number of G, denoted by γtRp(G), is the minimum weight of a TPRDF on G. In this paper, we initiated the study of total perfect Roman domination. We characterized graphs with the largest-possible γtRp(G). We proved that total perfect Roman domination is NP-complete for chordal graphs, bipartite graphs, and for planar bipartite graphs. Finally, we related γtRp(G) to perfect domination γp(G) by proving γtRp(G)≤3γp(G) for every graph G, and we characterized trees T of order n≥3 for which γtRp(T)=3γp(T). This notion can be utilized to develop a defensive strategy with some properties.

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Section: Introductionmentioning

confidence: 99%

Total Perfect Roman Domination

Almulhim

2023

Symmetry

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Perfect Roman {3}‐Domination in Graphs: Complexity and Bound of Perfect Roman {3}‐Domination Number of Trees

Almulhim

2024

Journal of Mathematics

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A perfect Roman {3}‐dominating function on a graph G = (V, E) is a function f : V⟶{0, 1, 2, 3} having the property that if f(v) = 0, then ∑u∈N(v)f(u) = 3, and if f(v) = 1, then ∑u∈N(v)f(u) = 2 for any vertex v ∈ V. The weight of a perfect Roman {3}‐dominating function f is the sum ∑v∈Vf(v). The perfect Roman {3}‐domination number of a graph G, denoted by , is the minimum weight of a perfect Roman {3}‐dominating function on G. In this paper, we initiate the study of a perfect Roman {3}‐domination, and we show that the decision problem associated with a perfect Roman {3}‐domination is NP‐complete for bipartite graphs. We also prove that if T is a tree of order n ≥ 2, then and characterize trees achieving this bound, and we give an infinity set of trees T of order n for which approaches this bound as n goes to infinity. Finally, we give the best upper bound of for some classes of graphs including regular, planar, and split graphs in terms of the order of the graphs.

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Vertex-edge perfect Roman domination number

Subaiei

Almulhim

Akwu

2023

MATH

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<abstract><p>A vertex-edge perfect Roman dominating function on a graph $ G = (V, E) $ (denoted by ve-PRDF) is a function $ f:V\left(G\right)\longrightarrow\{0, 1, 2\} $ such that for every edge $ uv\in E $, $ \max\{f(u), f(v)\}\neq0 $, or $ u $ is adjacent to exactly one neighbor $ w $ such that $ f(w) = 2 $, or $ v $ is adjacent to exactly one neighbor $ w $ such that $ f(w) = 2 $. The weight of a ve-PRDF on $ G $ is the sum $ w(f) = \sum_{v\in V}f(v) $. The vertex-edge perfect Roman domination number of $ G $ (denoted by $ \gamma_{veR}^{p}(G) $) is the minimum weight of a ve-PRDF on $ G $. In this paper, we first show that vertex-edge perfect Roman dominating is NP-complete for bipartite graphs. Also, for a tree $ T $, we give upper and lower bounds for $ \gamma_{veR}^{p}(T) $ in terms of the order $ n $, $ l $ leaves and $ s $ support vertices. Lastly, we determine $ \gamma_{veR}^{p}(G) $ for Petersen, cycle and Flower snark graphs.</p></abstract>

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The Perfect Roman Domination Number of the Cartesian Product of Some Graphs

Abstract: A perfect Roman dominating function on a graph G is a function f : V G ⟶ 0,1,2 … Show more

Cited by 3 publications

References 17 publications

Total Perfect Roman Domination

Total Perfect Roman Domination

Perfect Roman {3}‐Domination in Graphs: Complexity and Bound of Perfect Roman {3}‐Domination Number of Trees

Vertex-edge perfect Roman domination number

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