Outer-Independent Italian Domination in Graphs

An outer-independent k-rainbow dominating function of a graph G is a function f from V(G) to the set of all subsets of {1, 2,. .. , k} such that both the following hold: (i) {1,. .. , k} = u∈N(v) f (u) whenever v is a vertex with f (v) = ∅, and (ii) the set of all v ∈ V(G) with f (v) = ∅ is independent. The outer-independent k-rainbow domination number of G is the invariant γ k oir (G), which is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by an outer-independent k-rainbow dominating function. In this paper, we initiate the study of outer-independent k-rainbow domination. We first investigate the basic properties of the outerindependent k-rainbow domination and then we focus on the outer-independent 2-rainbow domination number and present sharp lower and upper bounds for it. ARTICLE HISTORY

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“…Fan et al [16] proved the following Nordhaus-Gaddum type result for outer-independent Italian domination number.…”

Section: Corollary 34mentioning

confidence: 95%

“…In [16], the authors proved that γ oiI (C n ) = n 2 for n ≥ 3 and γ oiI (K p,q ) = q for p ≥ q ≥ 2. Using these we obtain the next results.…”

Section: Corollary 33: Let G Be a Connected Graph Of Order N If G Hmentioning

confidence: 99%

Outer-independent k-rainbow domination

Kang

Samodivkin

Shao

et al. 2019

Journal of Taibah University for Science

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“…They also present the bound of γ I (C n 2P m ) for n, m ≥ 4. There are many studies on some domination numbers related to Italian domination, such as global Italian domination number [13], independent Italian domination number [14], outer-independent Italian domination number [15] and perfect Italian domination number [16].…”

Section: Introductionmentioning

confidence: 99%

More Results on Italian Domination in Cn□Cm

et al. 2020

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Italian domination can be described such that in an empire all cities/vertices should be stationed with at most two troops. Every city having no troops must be adjacent to at least two cities with one troop or at least one city with two troops. In such an assignment, the minimum number of troops is the Italian domination number of the empire/graph is denoted as γ I . Determining the Italian domination number of a graph is a very popular topic. Li et al. obtained γ I ( C n □ C 3 ) and γ I ( C n □ C 4 ) (weak {2}-domination number of Cartesian products of cycles, J. Comb. Optim. 35 (2018): 75–85). Stȩpień et al. obtained γ I ( C n □ C 5 ) = 2 n (2-Rainbow domination number of C n □ C 5 , Discret. Appl. Math. 170 (2014): 113–116). In this paper, we study the Italian domination number of the Cartesian products of cycles C n □ C m for m ≥ 6 . For n ≡ 0 ( mod 3 ) , m ≡ 0 ( mod 3 ) , we obtain γ I ( C n □ C m ) = m n 3 . For other C n □ C m , we present a bound of γ I ( C n □ C m ) . Since for n = 6 k , m = 3 l or n = 3 k , m = 6 l ( k , l ≥ 1 ) , γ r 2 ( C n □ C m ) = m n 3 , (the Cartesian product of cycles with small 2-rainbow domination number, J. Comb. Optim. 30 (2015): 668–674), it follows in this case that C n □ C m is an example of a graph class for which γ I = γ r 2 , which can partially answer the question presented by Brešar et al. on the 2-rainbow domination in graphs, Discret. Appl. Math. 155 (2007): 2394–2400.

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“…Zepeng Li et al [11] obtain the Italian domination numbers of C n 2C 3 and C n 2C 4 , where they call it weak {2}-domination number. Some scholars initiate other aspects of Italian domination, such as Abdelkader Rahmouni et al [12], who study independent Italian domination, Wenjie Fan et al [13], who study the outer-independent Italian domination number, and Lutz Volkmann [14], who initiates the study of the Italian domatic number in digraphs.…”

Section: Introductionmentioning

confidence: 99%

“…Zepeng Li et al [12] obtain the Italian domination numbers of C n 2C 3 and C n 2C 4 , where they call it weak {2}-domination number. Some scholars initiate other aspects of Italian domination, such as Abdelkader Rahmouni et al [13] study independent Italian domination, Wenjie…”

Section: Introductionmentioning

confidence: 99%

The Italian Domination Numbers of Generalized Petersen Graphs P(n,3)

Gao

et al. 2019

Mathematics

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An Italian dominating function of G is a function f : V ( G ) → { 0 , 1 , 2 } , for every vertex v such that f ( v ) = 0 , it holds that ∑ u ∈ N ( v ) f ( u ) ≥ 2 . The Italian domination number γ I ( G ) is the minimum weight of an Italian dominating function on G. In this paper, we determine the exact values of the Italian domination numbers of P ( n , 3 ) .

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Outer-Independent Italian Domination in Graphs

Cited by 14 publications

References 11 publications

Outer-independent k-rainbow domination

Outer-independent k-rainbow domination

More Results on Italian Domination in Cn□Cm

The Italian Domination Numbers of Generalized Petersen Graphs P(n,3)

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