A note on the independent Roman domination in unicyclic graphs

Chellali, Mustapha; Rad, Nader Jafari

doi:10.7494/opmath.2012.32.4.715

Cited by 15 publications

(9 citation statements)

References 5 publications

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“…Since at least two elements from V ∪ E are required to dominate any five consecutive vertices on a path, and these two elements dominate at most 5 consecutive edges, it is straightforward to check that γ * R (P n ) is at least 2 n 5 + i if n ≡ i (mod 5) for i ∈ {0, 1, 2, 4} and is at least 2 n 5 + 2 if n ≡ 3 (mod 5). Simplifying, we have that γ * R (P n ) is bounded below by 4n−2 5 + 1 if n ≡ 4 (mod 5) and 4n−2 5 , otherwise.…”

Section: Proposition 10 For a Non-trivial P N mentioning

confidence: 98%

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Mixed Roman Domination in Graphs

Ahangar

Haynes

Tripodoro

2015

Bull. Malays. Math. Sci. Soc.

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Let G = (V, E) be a simple graph with vertex set V and edge set E. A mixed Roman dominating functionThe mixed Roman domination number of G is the minimum weight of a mixed Roman dominating function of G. In this paper, we initiate the study of the mixed Roman domination number and we present bounds for this parameter. We characterize the graphs attaining an upper bound and the graphs having small mixed Roman domination numbers.

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Section: Proposition 10 For a Non-trivial P N mentioning

confidence: 98%

“…Roman domination was introduced by Cockayne et al [8] in 2004 and was inspired by the work of ReVelle and Rosing [15] and Stewart [16]. Since this introduction, Roman domination and its variations have been studied in the literature, see for example [1,[4][5][6]9,13].…”

Section: Introductionmentioning

confidence: 99%

Mixed Roman Domination in Graphs

Ahangar

Haynes

Tripodoro

2015

Bull. Malays. Math. Sci. Soc.

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“…1g is an independent set. The independent Roman domination number i R (G) is the minimum weight of an IRDF on G. The concept of independent Roman dominating function was first defined by Cockayne et al in [8] and has been studied by several authors [7,19].…”

Section: Introductionmentioning

confidence: 99%

Independent double Roman domination in graphs

Maimani

Momeni

Mahid

et al. 2020

AKCE International Journal of Graphs and Combinatorics

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For a graph G ¼ (V,E), a double Roman dominating function ðDRDFÞf : V ! f0, 1, 2, 3g has the property that for every vertex v 2 V with f(v) ¼ 0, either there exists a vertex u 2 NðvÞ, with f(u) ¼ 3, or at least two neighbors x, y 2 NðvÞ having f(x) ¼ f(y) ¼ 2, and every vertex with value 1 under f has at least a neighbor with value 2 or 3. The weight of a DRDF is the sum f ðVÞ ¼ P v2V f ðvÞ. A DRDF f is called independent if the set of vertices with positive weight under f, is an independent set. The independent double Roman domination number i dR ðGÞ is the minimum weight of an independent double Roman dominating function on G. In this paper, we show that for every graph G of order n, i r3 ðGÞ À i dR ðGÞ n=5 and iðGÞ þ i R ðGÞ À i dR ðGÞ n=4, where i r3 ðGÞ, i R ðGÞ and i(G) are the independent 3-rainbow domination, independent Roman domination and independent domination numbers, respectively. Moreover, we prove that for any tree G, i dR ðGÞ ! i r3 ðGÞ.

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“…Dominating theory provides a fancy tool to satisfy the safe requirements of many applications in security environments. The finite model would like to be controlled by lowest cost and energy [9]- [15], [17]. These types of networks are modeled as graphs.…”

Section: Introductionmentioning

confidence: 99%

Changing and Unchanging 2-Rainbow Independent Domination

Shi

Shao

et al. 2019

IEEE Access

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Domination number is of practical interest in several theoretical and applied scenes. In the problem of wireless networking, the dominating idea is used to deduce an efficient route within the adhoc mobilenetworks. It has also been used for the summarization of documents and making a design of secure web-systems for electrical recursive grids. In this paper, we explore an important class of domination numbers, which is the 2-rainbow independent dominating function (2RiDF) on graphs. The minimum weight of a 2RiDF on a graph G is called the 2-rainbow independent domination number of G. A graph G is 2-rainbow independent domination stable if the 2-rainbow independent domination number of G remains unchanged under removal of any vertex. As a result, we characterize 2-rainbow independent domination stable tree-networks and study the effect of edge removal on 2-rainbow independent domination number in trees. INDEX TERMS 2-rainbow independent domination number, 2-rainbow independent domination stable graph, tree.

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A note on the independent Roman domination in unicyclic graphs

Cited by 15 publications

References 5 publications

Mixed Roman Domination in Graphs

Mixed Roman Domination in Graphs

Independent double Roman domination in graphs

Changing and Unchanging 2-Rainbow Independent Domination

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