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2012 | 32 | 3 | 419-426
Tytuł artykułu

On the total k-domination number of graphs

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
32
Numer
3
Strony
419-426
Opis fizyczny
Daty
wydano
2012
otrzymano
2011-03-09
poprawiono
2011-07-23
zaakceptowano
2011-07-25
Twórcy
  • Department of Mathematics, University of Mohaghegh Ardabili, P.O. Box 5919911367, Ardabil, Iran
Bibliografia
  • [1] M. El-Zahar, S. Gravier and A. Klobucar, On the total domination number of cross products of graphs, Discrete Math. 308 (2008) 2025-2029, doi: 10.1016/j.disc.2007.04.034.
  • [2] F. Harary and T.W. Haynes, Double domination in graphs, Ars Combin. 55 (2000) 201-213.
  • [3] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
  • [4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs; Advanced Topics (Marcel Dekker, New York, 1998).
  • [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.
  • [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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1616
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