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Stępień, Zofia; Szymaszkiewicz, Alicja; Szymaszkiewicz, Lucjan; Zwierzchowski, Maciej

doi:10.1016/j.dam.2014.01.027

Cited by 14 publications

(8 citation statements)

References 10 publications

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“…3 [17], it follows in this case C n 2C m is an example of a graph class for which γ I = γ r2 . Thus, we give another example of a graph class for which γ I = γ r2 besides C n 2C 5 [10] and trees [6], partially answer the question of Brešar et al [9].…”

Section: Introductionsupporting

confidence: 57%

“…", and prove that for trees γ I (T) = γ r2 (T) [6]. Stȩpień et al [10] show γ I (C n 2C 5 ) = γ r2 (C n 2C 5 ). To determine the Italian domination number of a graph also attracts more scholars.…”

Section: Introductionmentioning

confidence: 96%

“…Li et al [11] give the value of γ I (C n 2C 3 ) and γ I (C n 2C 4 ). Stȩpień et al [10] obtain γ I (C n 2C 5 ) = 2n. Gao et al [12] determine the exact values of γ I (C n 2P 3 ) and γ I (C 3 2P m ).…”

Section: Introductionmentioning

confidence: 96%

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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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Section: Introductionsupporting

confidence: 57%

Section: Introductionmentioning

confidence: 96%

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More Results on Italian Domination in Cn□Cm

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“…Michael A. Henning et al [9] characterize the tree graphs T with γ(T) + 1 = γ I (T), and characterize T with γ I (T) = 2γ(T). Zofia Stȩpień et al [10] prove γ I (C n 2C 5 ) = 2n. 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.…”

Section: Introductionmentioning

confidence: 97%

“…demonstrate that the Italian domination number is equal to the 2-rainbow domination number for trees and cactus graphs with no even cycles. Michael A. Henning et al [10] characterize the tree graphs T with γ(T) + 1 = γ I (T), and characterize T with γ I (T) = 2γ(T). Zofia Stȩpień et al [11] prove γ I (C n 2C 5 ) = 2n.…”

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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Rainbow Domination in Graphs

Brešar

2020

Topics in Domination in Graphs

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2-Rainbow domination number ofCn□C5

Cited by 14 publications

References 10 publications

More Results on Italian Domination in Cn□Cm

More Results on Italian Domination in Cn□Cm

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

Rainbow Domination in Graphs

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