# On the total k-domination number of graphs

Discussiones Mathematicae Graph Theory (2012)

- Volume: 32, Issue: 3, page 419-426
- ISSN: 2083-5892

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topAdel P. Kazemi. "On the total k-domination number of graphs." Discussiones Mathematicae Graph Theory 32.3 (2012): 419-426. <http://eudml.org/doc/270955>.

@article{AdelP2012,

abstract = {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 τₖ(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.},

author = {Adel P. Kazemi},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {total k-domination (k-tuple total domination) number; k-tuple domination number; k-transversal number; total -domination number; -tuple total domination number; -tuple domination number; -transversal number},

language = {eng},

number = {3},

pages = {419-426},

title = {On the total k-domination number of graphs},

url = {http://eudml.org/doc/270955},

volume = {32},

year = {2012},

}

TY - JOUR

AU - Adel P. Kazemi

TI - On the total k-domination number of graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2012

VL - 32

IS - 3

SP - 419

EP - 426

AB - 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 τₖ(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.

LA - eng

KW - total k-domination (k-tuple total domination) number; k-tuple domination number; k-transversal number; total -domination number; -tuple total domination number; -tuple domination number; -transversal number

UR - http://eudml.org/doc/270955

ER -

## References

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- [2] F. Harary and T.W. Haynes, Double domination in graphs, Ars Combin. 55 (2000) 201-213. Zbl0993.05104
- [3] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). Zbl0890.05002
- [4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs; Advanced Topics (Marcel Dekker, New York, 1998). Zbl0883.00011
- [5] M.A. Henning and A.P. Kazemi, k-tuple total domination in graphs, Discrete Appl. Math. 158 (2010) 1006-1011, doi: 10.1016/j.dam.2010.01.009. Zbl1210.05097
- [6] M.A. Henning and A.P. Kazemi, k-tuple total domination in cross products of graphs, J. Comb. Optim. 2011, doi: 10.1007/s10878-011-9389-z.
- [7] V. Chvátal and C. Mc Diarmid, Small transversals in hypergraphs, Combinatorica 12 (1992) 19-26, doi: 10.1007/BF01191201. Zbl0776.05080

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