More Results on the Domination Number of Cartesian Product of Two Directed Cycles

Ye, Ansheng; Miao, Fang; Shao, Zehui; Liu, Jia‐Bao; Žerovnik, Janez; Repolusk, Polona

doi:10.3390/math7020210

Cited by 5 publications

(5 citation statements)

References 16 publications

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“…Recently, in 2019, Sivagami and Chelvam [17] considered the domination number of the trace graph of matrices. In addition, Ye et al [18] provided more results on the domination number of the Cartesian product of two directed cycles.…”

Section: Introductionmentioning

confidence: 99%

Domination parameters on Cayley digraphs of transformation semigroups with fixed sets

Nupo¹,

Pookpienlert²

2021

Turk J Math

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For a nonempty subset Y of a nonempty set X , denote by F ix(X, Y ) the semigroup of full transformations on the set X in which all elements in Y are fixed. The Cayley digraph Cay(F ix(X, Y ), A) of F ix(X, Y ) with respect to a connection set A ⊆ F ix(X, Y ) is defined as a digraph whose vertex set is F ix(X, Y ) and two vertices α, β are adjacent in sense of drawing a directed edge (arc) from α to β if there exists µ ∈ A such that β = αµ . In this paper, we determine domination parameters of Cay(F ix(X, Y ), A) where A is a subset of F ix(X, Y ) related to minimal idempotents and permutations in F ix(X, Y ).

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Section: Introductionmentioning

confidence: 99%

Domination parameters on Cayley digraphs of transformation semigroups with fixed sets

Nupo¹,

Pookpienlert²

2021

Turk J Math

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“…A set D ⊆ V(G) is a domination set if for each v of G either v ∈ D or v is adjacent to some u ∈ D. The minimum cardinality of dominating sets of G is the domination number, denoted by γ(G). Domination on graphs originates practical problems in Operations Research, so it has been extensively studied [1,2] and it has many variants, such as total domination [3], super domination [4], Roman domination [5], rainbow domination [6] and so on.…”

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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“…Vizing initiates the problem of domination on Cartesian product graphs [20]. Since then various domination numbers of G2H are extensively studied ( [21][22][23][24]).…”

Section: Introductionmentioning

confidence: 99%

The 3-Rainbow Domination Number of the Cartesian Product of Cycles

Gao

Yang

2020

Mathematics

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We have studied the k-rainbow domination number of C n □ C m for k ≥ 4 (Gao et al. 2019), in which we present the 3-rainbow domination number of C n □ C m , which should be bounded above by the four-rainbow domination number of C n □ C m . Therefore, we give a rough bound on the 3-rainbow domination number of C n □ C m . In this paper, we focus on the 3-rainbow domination number of the Cartesian product of cycles, C n □ C m . A 3-rainbow dominating function (3RDF) f on a given graph G is a mapping from the vertex set to the power set of three colors { 1 , 2 , 3 } in such a way that every vertex that is assigned to the empty set has all three colors in its neighborhood. The weight of a 3RDF on G is the value ω ( f ) = ∑ v ∈ V ( G ) | f ( v ) | . The 3-rainbow domination number, γ r 3 ( G ) , is the minimum weight among all weights of 3RDFs on G. In this paper, we determine exact values of the 3-rainbow domination number of C 3 □ C m and C 4 □ C m and present a tighter bound on the 3-rainbow domination number of C n □ C m for n ≥ 5 .

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More Results on the Domination Number of Cartesian Product of Two Directed Cycles

Cited by 5 publications

References 16 publications

Domination parameters on Cayley digraphs of transformation semigroups with fixed sets

Domination parameters on Cayley digraphs of transformation semigroups with fixed sets

More Results on Italian Domination in Cn□Cm

The 3-Rainbow Domination Number of the Cartesian Product of Cycles

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