<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-tuple total domination in graphs

Henning, Michael A.; Kazemi, Adel P.

doi:10.1016/j.dam.2010.01.009

Cited by 60 publications

(10 citation statements)

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“…After that, many variants of the dominating set were introduced. Among the most famous, we have the k-tuple total dominating set [18], which is an extension of the total dominating set. A set S of vertices in G is a k-tuple total dominating set, abbreviated kTD-set, of G if every vertex of G is adjacent to least k vertices in S. The minimum cardinality of a kTD-set of G is the k-tuple total domination number of G, denoted by γ ×k,t (G).…”

Section: Introductionmentioning

confidence: 99%

Double total dominator chromatic number of graphs

Beggas¹,

Kheddouci²,

Marweni³

2021

Discuss. Math. Graph Theory

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In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph G with minimum degree at least 2 is a proper vertex coloring of G such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of G is called the double total dominator chromatic number, denoted by χ t dd (G). Therefore, we establish the close relationship between the double total dominator chromatic number χ t dd (G) and the double total domination number γ ×2,t (G). We prove the NP-completeness of the problem. We also examine the effects on χ t dd (G) when G is modified by some operations. Finally, we discuss the χ t dd (G) number of square of trees by giving some bounds.

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Section: Introductionmentioning

confidence: 99%

Double total dominator chromatic number of graphs

Beggas¹,

Kheddouci²,

Marweni³

2021

Discuss. Math. Graph Theory

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“…Because of such a correspondence, many chemical and physical properties of molecules are in correlation with graph theoretical invariants. One very important such invariant is the total (double) domination number [2,[6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Total and Double Total Domination Number on Hexagonal Grid

Klobučar

2019

Mathematics

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“…In [5], Henning and Kazemi started the studying of the k-tuple total domination (= total k-domination) number in graphs. Let k be a positive integer.…”

Section: Introductionmentioning

confidence: 99%

“…The following theorems are useful in the context. Theorem 1 [5,6]. Let G be a graph with δ(G) ≥ k. Then for any integer m ≥ k+1, γ ×k,t (G) = m if and only if G = K ′ m or G = F • k K ′ m , for some graph F and some spanning subgraph K ′ m of K m such that K ′ m has minimum degree at least k and m is minimum in the set {t :…”

Section: Introductionmentioning

confidence: 99%

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On the total k-domination number of graphs

Kazemi

2012

Discuss. Math. Graph Theory

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Let k be a positive integer and let G = (V, E) be a simple graph. The k-tuple domination number γ ×k (G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V , |N G [v] ∩ S| ≥ k. Also the total k-domination number γ ×k,t (G) of G is the minimum cardinality of a total k-dominating set S, a set that for every vertex v ∈ V , |N G (v) ∩ S| ≥ k. The k-transversal number τ k (H) of a hypergraph H is the minimum size of a subset S ⊆ V (H) such that |S ∩ e| ≥ k for every edge e ∈ E(H). We know that for any graph G of order n with minimum degree at least k, γ ×k (G) ≤ γ ×k,t (G) ≤ n. Obviously for every k-regular graph, the upper bound n is sharp. Here, we give a sufficient condition for γ ×k,t (G) < n. Then we characterize complete multipartite graphs G with γ ×k (G) = γ ×k,t (G). We also state that the total k-domination number of a graph is the k-transversal number of its open neighborhood hypergraph, and also the domination number of a graph is the transversal number of its closed neighborhood hypergraph. Finally, we give an upper bound for the total k-domination number of the cross product graph G × H of two graphs G and H in terms on the similar numbers of G and H. Also, we show that this upper bound is strict for some graphs, when k = 1.

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k-tuple total domination in graphs

Cited by 60 publications

References 5 publications

Double total dominator chromatic number of graphs

Double total dominator chromatic number of graphs

Total and Double Total Domination Number on Hexagonal Grid

On the total k-domination number of graphs

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