Double Roman Domination in Digraphs

Note: Please see pdf for full abstract with equations. Let D = (V,A) be a digraph. A double Roman dominating function on a digraph D is a function ƒ :V → {0, 1, 2, 3} such that every vertex u for which ƒ(u) = 0 has an in-neighbor v for which ƒ(v) = 3 or at least two in-neighbors assigned 2 under ƒ, while if ƒ(u) = 1, then the vertex u must have at least one in-neighbor assigned 2 or 3. The weight of a double Roman dominating function is the value ƒ(V) = ∑v∈V ƒ(v). The minimum weight of a double Roman dominating function on a digraph D is called the double Roman domination number of D, denoted by γdR (D). This paper gives a descriptive characterization of oriented trees T satisfying γdR(T ) = 2 (n − Δ+) + 1. 2020 Mathematics Subject Classification: 05C20,05C69.

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“…Proposition 5 [3] For any digraph D, there exists a dR -function such that no vertex needs to be assigned the value 1.…”

Section: Resultsmentioning

confidence: 99%

“…For digraphs, the de…nition of double Roman dominating function was introduced by Hao et al [3], and some results were given. In [6], we gave extremal k-out-regular digraphs with 1 k n 1 and tournaments, atteining the next upper bound, given in [3].…”

Section: Introductionmentioning

confidence: 99%

On the double Roman domination number in oriented trees

Ouldrabah

Bouchou

Blidia

2023

Preprint

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“…Specifically, let D be a finite digraph with neither loops nor multiple arcs ( In this paper we continue the study of double Roman dominating functions and double Roman domatic numbers in graphs and digraphs (see, for example, [1-5, 7, 9, 11]). Inspired by an idea of the work [4], we defined in [5]…”

Section: Terminology and Introductionmentioning

confidence: 99%

“…According to [5], we can assume, without loss of generality, that no vertex of f i is assigned the value 1. In [5], the authors show this for γ dR (D)-functions, however, the same proof works for each DRD function. Since ∆ + (D) ≤ (n − k)/(k − 1), we observe that f i (x) ≥ 2 for at least k different vertices for each i ∈ {1, 2, .…”

Section: Terminology and Introductionmentioning

confidence: 99%

The double Roman domatic number of a digraph

Volkmann¹

2020

Discuss. Math. Graph Theory

Self Cite

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A double Roman dominating function on a digraph D with vertex set V (D) is defined in [G. Hao, X. Chen and L. Volkmann, Double Roman domination in digraphs, Bull. Malays. Math. Sci. Soc. (2017).] as a function f : V (D) → {0, 1, 2, 3} having the property that if f (v) = 0, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor w with f (w) = 3, and if f (v) = 1, then the vertex v must have at least one in-neighbor u with f (u) ≥ 2. A set {f 1 , f 2 ,. .. , f d } of distinct double Roman dominating functions on D with the property that d i=1 f i (v) ≤ 3 for each v ∈ V (D) is called a double Roman dominating family (of functions) on D. The maximum number of functions in a double Roman dominating family on D is the double Roman domatic number of D, denoted by d dR (D). We initiate the study of the double Roman domatic number, and we present different sharp bounds on d dR (D). In addition, we determine the double Roman domatic number of some classes of digraphs.

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“…Yue et al [29] obtained that a graph with order n ≥ 3 satisfies 8 ≤ γ dR (G) + γ dR (G) ≤ 2n + 3 and characterized the graphs attaining these bounds. For other results on γ dR (G), we refer to [5,21,24,25,28].…”

Section: Introductionmentioning

confidence: 99%

Integer linear programming formulations for double roman domination problem

Cai

Fan

Shi

et al. 2019

Optimization Methods and Software

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For a graph G = (V, E), a double Roman dominating function (DRDF) is a function f : V → {0, 1, 2, 3} having the property that if f (v) = 0, then vertex v must have at least two neighbors assigned 2 under f or one neighbor u with f (u) = 3, and if f (v) = 1, then vertex v must have at least one neighbor u with f (u) ≥ 2. In this paper, we consider the double Roman domination problem (DRDP), which is an optimization problem of finding the DRDF f such that v∈V f (v) is minimum. We propose several integer linear programming (ILP) formulations with a polynomial number of constraints for this combinatorial optimization problem, and present computational results for graphs of different types and sizes. Further, we prove that some of our ILP formulations are equivalent to the others regardless of the variables relaxation or usage of less number of constraints and variables. Additionally, we use one ILP formulation to give an H(2(∆ + 1))-approximation algorithm and provide an in approximability result for this problem. All proposed ILP formulations and approximation algorithm are evaluated on randomly generated graphs to compare the performance.

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Double Roman Domination in Digraphs

Cited by 14 publications

References 12 publications

On the double Roman domination number in oriented trees

On the double Roman domination number in oriented trees

The double Roman domatic number of a digraph

Integer linear programming formulations for double roman domination problem

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